Transform coding of signals using approximate trigonometric expansions

For signal representation, it is always preferred that a signal be represented using a minimum number of parameters. In any transform coding scheme, the central operation is the reduction of correlation and thereby, with appropriate coding of the transform coefficients, allows data compression to be achieved. The objective of data encoding is to transform a data array into a statistically uncorrelated set. This step is typically considered a "decorrelation" step, because in the case of unitary transformations, the resulting transform coefficients are relatively uncorrelated. Most unitary transforms have the tendency to compact the signal energy into relatively few coefficients. The compaction of energy thus achieved permits a prioritization of the spectral coefficients, with the most energetic ones receiving a greater allocation of encoding bits. The transform efficiency and ease of implementation are to a large extent mutually incompatible. There are various transforms such as Karhunen-Loeve, discrete cosine transforms, etc., but the choice depends on the amount of reconstruction error that can be tolerated and the computational resources available. We apply an approximate Fourier series expansion (AFE) to sampled one-dimensional signals and images, and investigate some mathematical properties. Additionally, we extend the expansion to an approximate cosine expansion (ACE) and show that, for the purpose of data compression with minimum error reconstruction of images, the performance of ACE is better than AFE. For comparison purposes, the results are also compared with a discrete cosine transform (DCT).