Discrete Cosine Transform

A discrete cosine transform (DCT) expresses a finite sequence of data points in terms of a sum of cosine functions oscillating at different frequencies. DCTs are important to numerous applications in science and engineering, from lossy compression of audio (e.g. MP3) and images (e.g. JPEG) (where small high-frequency components can be discarded) to spectral methods for the numerical solution of partial differential equations. The use of cosine rather than sine functions is critical for compression since it turns out (as described below) that fewer cosine functions are needed to approximate a typical signal, whereas for differential equations the cosines express a particular choice of boundary conditions.

In particular, a DCT is a Fourier-related transform similar to the discrete Fourier transform (DFT), but using only real numbers. The DCTs are generally related to Fourier Series coefficients of a periodically and symmetrically extended sequence whereas DFTs are related to Fourier Series coefficients of a periodically extended sequence. DCTs are equivalent to DFTs of roughly twice the length, operating on real data with even symmetry (since the Fourier transform of a real and even function is real and even), whereas in some variants the input and/or output data are shifted by half a sample. There are eight standard DCT variants, of which four are common.

Haar Transform

The Haar Transform, or the Haar Wavelet Transform (HWT) is one of a group of related transforms known as the Discrete Wavelet Transforms (DWT). DWT Transforms, and the Haar transform in particular can frequently be made very fast using matrix calculations. The fastest known algorithm for computing the HWT is known as the Fast Haar Transform, and is comparable in speed and properties to the Fast Fourier Transform.

Uses of the Haar Transform

Haar transform uses non-sinusoidal basic wavefunction. So it has great applications related to DSP. The basic haar transform matrix is defined by the function Hk(x). Where o<=k<=N-1, N is the matrix size.