 systems and indeed in 2G systems at the application layer (layer 7) to source code voice, audio, image and video. Almost all compression algorithms use an FFT at some stage to achieve bandwidth efficiency, exploiting the ability of the Fourier transform to represent complex signals viewed in the frequency domain as a composite of simple sinusoidal signals. In 4G radio systems we user Fourier transforms at the physical layer (layer1) to translate a symbol stream on to discrete frequency subcarriers. The effect is to provide additional frequency diversity and time diversity in that the symbols are further apart in time than the original bit stream. We do this SO that we can support higher single-user data rates. However, present cellular systems make wide use of the Hadamard transform both in the channel encoding/decoding process (convolution and block coding) and code division multiple access (CDMA). Channel coding provides time diversity. CDMA provides frequency diversity (by spreading the original data signal across a wider bandwidth). Does Fourier really deliver significant additional performance gain? To answer this we need to consider the history of matrix maths. 3.2.1 The Origins of Matrix Maths and the Hadamard Transform Matrix maths is the science of putting numbers in boxes or rather, arranging numbers in rows and columns (a horizontal and vertical matrix). The numbers can be binary or nonbinary. The Sumerian abacus came into use around 2700 to 2300 BC providing a mechanism for tabling successive columns of successive orders of magnitude. The Babylonians in the third or fourth century BC used matrices on clay tablets, a forerunner of the counting tables used by the Romans. Counting tables were simply tables with a ridged edge. The tables contained sand and the sand would be divided up into squares to help in counting and calculation. Matrix concepts were developed by the Chinese in the Han Dynasty between 200 and 100 BC to solve linear equations. This implied an understanding that matrices exhibit certai