Discrete Cosine Transform Thesis

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In a more subtle fashion, the boundary conditions are responsible for the "energy compactification" properties that make DCTs useful for image and audio compression, because the boundaries affect the rate of convergence of any Fourier-like series.

In particular, it is well known that any discontinuities in a function reduce the rate of convergence of the Fourier series, so that more sinusoids are needed to represent the function with a given accuracy.

DCT-IV has gained popularity for its applications in fast implementation of real-valued polyphase filtering banks, A variant, the modified discrete cosine transform, or MDCT (based on the DCT-IV), is used in the MP3, AAC, Vorbis and WMA audio compression formats.

Like any Fourier-related transform, discrete cosine transforms (DCTs) express a function or a signal in terms of a sum of sinusoids with different frequencies and amplitudes.

Like the discrete Fourier transform (DFT), a DCT operates on a function at a finite number of discrete data points.

The obvious distinction between a DCT and a DFT is that the former uses only cosine functions, while the latter uses both cosines and sines (in the form of complex exponentials).

Multidimensional DCTs (MD DCTs) are developed to extend the concept of DCT on MD Signals. A new variety of fast algorithms are also developed to reduce the computational complexity of implementing DCT. DCTs are also widely employed in solving partial differential equations by spectral methods, where the different variants of the DCT correspond to slightly different even/odd boundary conditions at the two ends of the array.

The discrete cosine transform (DCT) was first conceived by Nasir Ahmed while working at the University of Texas, and he proposed the concept to the National Science Foundation in 1972. DCTs are also closely related to Chebyshev polynomials, and fast DCT algorithms (below) are used in Chebyshev approximation of arbitrary functions by series of Chebyshev polynomials, for example in Clenshaw–Curtis quadrature.

(A similar problem arises for the DST, in which the odd left boundary condition implies a discontinuity for any function that does not happen to be zero at that boundary.) In contrast, a DCT where both boundaries are even always yields a continuous extension at the boundaries (although the slope is generally discontinuous).

This is why DCTs, and in particular DCTs of types I, II, V, and VI (the types that have two even boundaries) generally perform better for signal compression than DFTs and DSTs.

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