The DiscreteTransforms package contains commands for computing transforms of discrete data.

The commands of the package are (currently) limited to hardware precision, which means that the results are independent of Digits, and that all computations occur in double precision (approximately 15 decimal digits).

The Fourier commands in the package work with almost any compatible data type, but are most efficient when used with the natural data type that is used to compute the transform. The wavelet commands in the package work purely with real, float[8] data.

Each command in the DiscreteTransforms package can be accessed by using either the long form or the short form of the command name in the command calling sequence.

The long form, DiscreteTransforms:-command, is always available. The short form can be used after loading the package.

The following is a list of available commands:

For information about efficient use of the FourierTransform and InverseFourierTransform commands, see FourierTransform/efficiency.

To display the help page for a particular DiscreteTransforms command, see Getting Help with a Command in a Package.

$\mathrm{with}\left(\mathrm{DiscreteTransforms}\right)$

$\left[\mathrm{DiscreteWaveletTransform}\,\mathrm{FourierTransform}\,\mathrm{InverseDiscreteWaveletTransform}\,\mathrm{InverseFourierTransform}\,\mathrm{WaveletCoefficients}\,\mathrm{WaveletPlot}\right]$

$Z≔\mathrm{Vector}\left(4\,i\mapsto \mathrm{evalf}\left[15\right]\left(1+\cos \left(\frac{\mathrm{\pi }\cdot i}{2}\right)+\mathrm{I}\cdot \sin \left(\frac{\mathrm{\pi }\cdot i}{2}\right)\right)\,\mathrm{datatype}=\mathrm{complex}\left[8\right]\right)$

$Z≔\left[\begin{array}{c}1.+\mathrm{I}\\ 0.+0.\mathrm{I}\\ 1.-\mathrm{I}\\ 2.+0.\mathrm{I}\end{array}\right]$

$\mathrm{FourierTransform}\left(Z\,\mathrm{inplace}=\mathrm{true}\right)$

$\left[\begin{array}{c}2.+0.\mathrm{I}\\ 0.+2.\mathrm{I}\\ 0.+0.\mathrm{I}\\ 0.+0.\mathrm{I}\end{array}\right]$

$\mathrm{InverseFourierTransform}\left(Z\,\mathrm{inplace}=\mathrm{true}\right)$

$\left[\begin{array}{c}1.+\mathrm{I}\\ 0.+0.\mathrm{I}\\ 1.-\mathrm{I}\\ 2.+0.\mathrm{I}\end{array}\right]$

$\mathrm{w1},\mathrm{w2}≔\mathrm{WaveletCoefficients}\left(\mathrm{Daubechies}\,4\right)$

$\mathrm{w1},\mathrm{w2}≔\left[\begin{array}{c}\mathrm{-0.129409522551260}\\ \mathrm{-0.224143868042013}\\ 0.836516303737808\\ \mathrm{-0.482962913144534}\end{array}\right],\left[\begin{array}{c}0.482962913144534\\ 0.836516303737808\\ 0.224143868042013\\ \mathrm{-0.129409522551260}\end{array}\right]$

$V≔\mathrm{Vector}\left[\mathrm{row}\right]\left(10\,i\mapsto i\,\mathrm{datatype}=\mathrm{float}\left[8\right]\right)$

$V≔\left[\begin{array}{cccccccccc}1.& 2.& 3.& 4.& 5.& 6.& 7.& 8.& 9.& 10.\end{array}\right]$

$\mathrm{V1},\mathrm{V2}≔\mathrm{DiscreteWaveletTransform}\left(V\,\mathrm{w1}\,\mathrm{w2}\right)$

$\mathrm{V1},\mathrm{V2}≔\left[\begin{array}{ccccc}2.22044604925031\times {10}^{\mathrm{-16}}& 4.44089209850063\times {10}^{\mathrm{-16}}& 4.44089209850063\times {10}^{\mathrm{-16}}& 8.88178419700125\times {10}^{\mathrm{-16}}& \mathrm{-3.53553390593274}\end{array}\right],\left[\begin{array}{ccccc}2.31078903454115& 5.13921615928734& 7.96764328403353& 10.7960704087797& 12.6771540786184\end{array}\right]$

$\mathrm{InverseDiscreteWaveletTransform}\left(\mathrm{V1}\,\mathrm{V2}\,\mathrm{w1}\,\mathrm{w2}\right)$

$\left[\begin{array}{cccccccccc}0.999999999999999& 2.00000000000000& 3.& 4.& 5.& 6.00000000000000& 7.& 8.00000000000000& 9.& 10.\end{array}\right]$