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LinearAlgebra`FourierTrig`

The discrete sine and cosine transforms are real variants of the discrete Fourier transform which are widely applicable from signal processing to the numerical solution of partial differential equations.

Discrete transforms.

There are several different conventions used to define these. In Mathematica, the discrete cosine transform of of a list of length is taken to be

and the discrete sine transform of a list of length is taken to be

This loads the package.

In[1]:= << LinearAlgebra`FourierTrig`

Here is some example data.

In[2]:= Table[1 - 2 Abs[x - .5], {x, .1, .9, .1}]

Out[2]=

Here is the discrete sine transform of the data.

In[3]:= Chop[FourierSin[%]]

Out[3]=

If this were the initial profile of, for example, a plucked string attached at both ends, this shows that only the odd modes (harmonics) are excited.

Mathematica has normalized the discrete sine and cosine transforms so that they are each their own inverses.

In[4]:= FourierSin[%]

Out[4]=