Built-in Mathematica Symbol

FourierDCT

FourierDCT[list]
finds the Fourier discrete cosine transform of a list of real numbers.

FourierDCT[list, m]
finds the Fourier discrete cosine transform of type

.

MORE INFORMATION

Possible types of discrete cosine transform for a list of length giving a result are:

1. DCT-I

2. DCT-II

3. DCT-III

4. DCT-IV

FourierDCT[list] is equivalent to FourierDCT[list, 2].

The inverse discrete cosine transforms for types 1, 2, 3, 4 are types 1, 3, 2, 4, respectively.

The list given in FourierDCT[list] can be nested to represent an array of data in any number of dimensions.

The array of data must be rectangular.

If the elements of list are exact numbers, FourierDCT begins by applying N to them.

FourierDCT can be used on SparseArray objects.

EXAMPLES

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Basic Examples (2)

Find a discrete cosine transform:

Find the inverse discrete cosine transform:

Find a discrete cosine transform of type 1 (DCT-I):

Find the inverse discrete cosine transform:

Find a discrete cosine transform:

In[1]:=

Out[1]=

Find the inverse discrete cosine transform:

In[2]:=

Out[2]=

Find a discrete cosine transform of type 1 (DCT-I):

In[1]:=

Out[1]=

Find the inverse discrete cosine transform:

In[2]:=

Out[2]=

Scope (2)

Use machine arithmetic to compute the discrete cosine transform:

Use 24-digit precision arithmetic:

A two-dimensional discrete cosine transform:

A five-dimensional discrete cosine transform:

Generalizations & Extensions (2)

The list may have complex values:

You can use "I", "II", "III", or "IV" for the types 1, 2, 3, and 4 respectively.

Applications (3)

Import some image data:

The two-dimensional DCT:

The diagonal spectra shows exponential decay:

Truncate modes in each axis, effectively compressing by factor of

:

Invert the DCT:

Get an expansion for an even function as a sum of cosines:

The function values on a uniformly spaced grid with n points on [-L,L):

Compute the DCT-III and renormalize:

The function has, in effect, been periodized with a particular symmetry:

Plot the expansion error where the points are defined:

Get an expansion for a function in the Chebyshev polynomials:

The values of the function at the Chebyshev nodes:

Find the Chebyshev coefficients:

Show the error:

Properties & Relations (3)

DCT-I and DCT-IV are their own inverses:

DCT-II and DCT-III are inverses of each other:

The DCT is equivalent to matrix multiplication:

Possible Issues (1)

FourierDCT always returns normalized results:

To get unnormalized results, you can multiply by the normalization: