DCTIV - Discrete Consine Transform type IV

Usage

c=dctiv(f);

Description

dctiv(f) computes the discrete cosine transform of type IV of the input signal f. If f is multi-dimensional, the transformation is applied along the first non-singleton dimension.

dctiv(f,L) zero-pads or truncates f to length L before doing the transformation.

dctiv(f,[],dim) or dctiv(f,L,dim) applies the transformation along dimension dim.

The transform is real (output is real if input is real) and orthonormal. It is its own inverse.

Let f be a signal of length L and let c=dctiv(f). Then

\begin{equation*} c\left(n+1\right)=\sqrt{\frac{2}{L}}\sum_{m=0}^{L-1}f\left(m+1\right)\cos\left(\frac{\pi}{L}\left(n+\frac{1}{2}\right)\left(m+\frac{1}{2}\right)\right) \end{equation*}

Examples:

The following figures show the first 4 basis functions of the DCTIV of length 20:

% The dctiv is its own adjoint.
F=dctiv(eye(20));

for ii=1:4
  subplot(4,1,ii);
  stem(F(:,ii));
end;

References:

K. Rao and P. Yip. Discrete Cosine Transform, Algorithms, Advantages, Applications. Academic Press, 1990.

M. V. Wickerhauser. Adapted wavelet analysis from theory to software. Wellesley-Cambridge Press, Wellesley, MA, 1994.