\(\text{DCT}_2\)

Calculates the type 2 discrete cosine transform, of a matrix in one or two dimensions.

You can use the \dct2 backslash command to insert this function.

The following variants of this function are available:

• \(\text{real matrix } \text{DCT}_2 \left ( \text{<matrix>} \right )\)

The \(\text{DCT}_2\) accepts boolean, integer, or real matrices and returns a real matrix containing the type 2 or forward DCT. The input matrix can be a row matrix, column matrix or a 2 dimensional matrix. The returned matrix will have the same dimensions as the supplied input matrix.

This function calculates the DFT in one dimension using the relation:

\[\begin{split}C _ k = \begin{cases} \frac{1}{\sqrt{N}} & k = 1 \\ \frac{\sqrt{2}}{\sqrt{N}} & k \ne 1 \\ \end{cases}\end{split}\]

\[y_k = C _ k \sum_{n=1}^N x_n \cos \frac{\left ( 2 n - 1 \right ) \pi \left ( k - 1 \right ) } {2N}\]

\[\forall k \in \left \lbrace 1 \ldots N \right \rbrace\]

The two dimensional DCT relies on the property of orthogonality. A DCT is first calculated independently on each row, creating a partially transformed matrix. A DCT is then calculated on each column of the partially transformed matrix.

Figure 122 shows the basic use of the \(\text{DFT}_2\) function.

Figure 122 Example Use of The Type 2 DCT Function