pplied maths) helps us to do things we want and need to do. The real-life signals that need to be transformed in communications all start out and end as analogue waveforms 26 LTE for UMTS Evolution to LTE Advanced, Second Edition, John Wiley, ISBN 978-0-470-66000-3. Making Telecoms Work (sound waves, light waves and radio waves) SO an understanding of waveform geometry as a specific part of mathematical theory is particularly important to us. Information in radio systems is modulated onto radio waves. The radio waves are sinusoidal but carry information that is modulated by changing the phase, frequency and/or amplitude of the (composite) carrier. We express phase, amplitude and frequency in terms of radians speed and angle of rotation of the phase vector). These waveforms can be described mathe- matically and manipulated mathematically. Mathematical manipulation and the efficiency of the mathematical manipulation process are the basis for modern digital radio communication systems. The history of maths and the contribution that individual mathematicians have made to waveform and transform theory goes something like this: 15 000 BC - the start of organised agriculture involves the counting of goats and other mobile objects of value. This is arithmetic, the science of numbers. Farmers need to work out the size of their fields. This is geometry - the science of shapes, from the Greek geo meaning earth and metro to measure. Then structures need to be built. This requires trigonometry - the science of angles. The amount of land and the fertility of the land available determines how many goats you can support. 2000 BC - the Egyptians produce a numeral system to support trade and agriculture 1900-1600 BC - the Babylonians introduce the base 60 system of applied mathe- matics still used today with 360 degrees in a circle and 60 minutes in an hour and 60 seconds in a minute. 640-546 BC - Thales of Miletus predicts the lunar eclipse of 585 BC and proves a number of properties of triangles including vertical angles being e