 Making Telecoms Work and combining them to create complex signals in the time domain, the inverse discrete Fourier transform used in OFDM signal synthesis and decomposing com- plex time-domain signals into simple frequency components, the discrete Fourier transform used in OFDM signal analysis. Unfortunately the process involves a lot of calculations and proves to be laborious. The theory is understood, the practice is impractical. 1777-1855 - Carl Friedrich Gauss develops the theory of complex numbers fol- lowing on from Euler and works out simpler and faster ways of doing Fourier transforms. He forgets to tell anyone how it's done. His work on the fast Fourier transform is published after his death in neo-Latin and remains unread by anyone who could possibly understand it. He also produces a body of work on non- Euclidean geometry, triangles with more than 180 degrees in their angles, and work on statistics (Gaussian distribution) which was to prove fundamental to our later understanding of white noise and filter design. 1845-1918 - Georg Cantor works on exotic number theory and transcendental and irrational numbers. (A number is transcendental if it is not the solution of any polynomial equation with integer coefficients. A number is irrational if there is no limit to the number of its decimal places. Pi and Euler's constant are examples of irrational numbers though bizarrely when added together they become rational.) 1887-1920 - Srinivasa Ramanujan produces ever more accurate approximations of Pi. 1862-1943 - David Hilbert works on algebraic number theory and geometry including the functional analysis of space and distance, the concept of vectors as linear sequences of complex numbers and functions and related work on harmonic and functional analysis. Hilbert is rather overshadowed by Einstein (1879-1955) but his contributions to modern communications, signal processing and vector theory remain very significant. He has a transform named after him, the Hilbert transform used in wideband 90 degree phase shifts