The French scientist Joseph B. Fourier developed the first important development in the theory of heat conduction presented to The Academy of Sciences in Paris in 1807. Fourier became a member of the Academy of Sciences in 1817. He continued to develop his ideas and eventually authored the applied mathematics classic Theorie Analytique de la Chaleur in 1822. Synchronized Fourier transforms are most often found in “order-tracking” for vibration analysis of rotating equipment such as engines, motor/generators, turbines, fans, and so on. Early on in the history of digital signal processing it was recognized that many of the multiplies in a discrete Fourier transform are actually redundant, making a fast Fourier transform possible through computer algorithm optimization. The input buffer of the Fourier transform can be seen as the product of the data window times an “infinitely long” waveform of the input signal.