Discrete Cosine Transform

Usage

dct(x, norm_forward = TRUE)

A vector or matrix the same size as x containing the Discrete Cosine Transform.

The DCT definitions here are based on the python scipy.fft.dct definitions. Specifically this use:

# python code
scipy.fft.dct(x, orthogonalize = True)

When norm_forward = TRUE

$$ y_k = \frac{1}{zN} \sum_{j=0}^{N-1}x_j\cos\left(\frac{\pi k(2j+1)}{2N}\right) $$

$$ z = \begin{cases} \sqrt{2}& \text{for }k=0\\ 1 & \text{for }k>0 \end{cases} $$

When norm_forward = FALSE $$ y_k = \frac{2}{z} \sum_{j=0}^{N-1}x_j\cos\left(\frac{\pi k(2j+1)}{2N}\right) $$

$$ z = \begin{cases} \sqrt{2}& \text{for }k=0\\ 1 & \text{for }k>0 \end{cases} $$

This second formulation is primarily to be able to generate the DCT basis functions like so

dct(diag(10), norm_forward = FALSE)

For the Inverse Discrete Cosine Transform, see idct.

x <- seq(0,1, length = 10)
y <- 5 + x + (2 * (x^2)) + (-2 * (x^4))

dct_coefs <- dct(y)