Approximate Fourier Expansion: Its Properties And Applications

For signal representation, it is always desired that a signal be represented using minimum number of parameters. An important criterion of signal representation is the orthogonality of the constituent basis functions of a transform. There are various orthogonal transforms such like Karhunen-Loeve, discrete cosine, Haar, discrete Fourier etc., but the choice of a particular transform in a given application depends on the amount of reconstruction error that can be tolerated and the computational resources available. The approximate Fourier expansion (AFE) for non-periodic signals with theoretically uncorrelated coefficients has previously been proposed. In this paper, we will give system interpretation to approximate Fourier expansion (AFE) using generalized harmonic analysis. Furthermore, we will investigate some mathematical properties of discrete approximate Fourier expansion (AFE). Finally, we will apply approximate Fourier expansion to images, and show that for purposes of decorrelation of transform coefficients and minimum error reconstruction of images, its performance is better than discrete Fourier transform (DFT). For comparison purposes, the results will also be compared with discrete cosine transform (DCT). Computer simulation results will also be presented