# FourierDCTMatrix

returns an n×n discrete cosine transform matrix of type 2.

FourierDCTMatrix[n,m]

returns an n×n discrete cosine transform matrix of type m.

# Details and Options

• Each entry Frs of the discrete cosine transform matrix of type m is computed as:
•  1 DCT-I 2 DCT-II 3 DCT-III 4 DCT-IV
• The discrete cosine transform matrices of types 1, 2, 3 and 4 have inverses of type 1, 3, 2 and 4, respectively. »
• Rows of the FourierDCTMatrix are basis sequences of the discrete cosine transform.
• The result of FourierDCTMatrix[n].list is equivalent to FourierDCT[list] when list has length n. However, the computation of FourierDCT[list] is much faster and has less numerical error. »
• For type 4, the option TargetStructure is supported, which specifies the structure of the returned matrix. Possible settings for TargetStructure include:
•  Automatic automatically choose the representation returned "Dense" represent the matrix as a dense matrix "Hermitian" represent the matrix as a Hermitian matrix "Orthogonal" represent the matrix as an orthogonal matrix "Symmetric" represent the matrix as a symmetric matrix "Unitary" represent the matrix as a unitary matrix
• is equivalent to FourierDCTMatrix[,TargetStructure"Dense"].
• gives a matrix with entries of precision p.

# Examples

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## Basic Examples(1)

A 4×4 DCT matrix:

## Scope(1)

The discrete cosine transform's basis sequences of length 128:

## Options(2)

### TargetStructure(1)

Return the DCT matrix as a dense matrix:

Return the DCT matrix as an orthogonal matrix:

Return the DCT matrix as a symmetric matrix:

### WorkingPrecision(1)

Use machine precision:

Use arbitrary precision:

## Applications(1)

Define 2D discrete cosine transform of size 8×8 using matrix formulation:

Simplified JPEG compression algorithm:

Compare the original and compressed images:

## Properties & Relations(2)

A DCT matrix multiplied by a vector is equivalent to the discrete cosine transform of that vector:

FourierDCT is much faster than the matrix-based computation:

A discrete cosine transform matrix of type 1 is its own inverse:

A discrete cosine transform matrix of type 3 is an inverse of the type 2 matrix:

A discrete cosine transform matrix of type 4 is its own inverse:

Wolfram Research (2012), FourierDCTMatrix, Wolfram Language function, https://reference.wolfram.com/language/ref/FourierDCTMatrix.html (updated 2024).

#### Text

Wolfram Research (2012), FourierDCTMatrix, Wolfram Language function, https://reference.wolfram.com/language/ref/FourierDCTMatrix.html (updated 2024).

#### CMS

Wolfram Language. 2012. "FourierDCTMatrix." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2024. https://reference.wolfram.com/language/ref/FourierDCTMatrix.html.

#### APA

Wolfram Language. (2012). FourierDCTMatrix. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/FourierDCTMatrix.html

#### BibTeX

@misc{reference.wolfram_2024_fourierdctmatrix, author="Wolfram Research", title="{FourierDCTMatrix}", year="2024", howpublished="\url{https://reference.wolfram.com/language/ref/FourierDCTMatrix.html}", note=[Accessed: 05-August-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_fourierdctmatrix, organization={Wolfram Research}, title={FourierDCTMatrix}, year={2024}, url={https://reference.wolfram.com/language/ref/FourierDCTMatrix.html}, note=[Accessed: 05-August-2024 ]}