# Fourier Transforms: An Introduction for Engineers by Robert M. Gray

By Robert M. Gray

The Fourier rework is likely one of the most vital mathematical instruments in a large choice of fields in technology and engineering. within the summary it may be considered because the transformation of a sign in a single area (typically time or house) into one other area, the frequency area. purposes of Fourier transforms, referred to as Fourier research or harmonic research, offer helpful decompositions of signs into basic or "primitive" parts, offer shortcuts to the computation of advanced sums and integrals, and sometimes show hidden constitution in info. Fourier research lies on the base of many theories of technological know-how and performs a basic function in functional engineering layout. The origins of Fourier research in technological know-how are available in Ptolemy's decomposing celestial orbits into cycles and epicycles and Pythagorus' de­ composing track into consonances. Its smooth background begun with the eighteenth century paintings of Bernoulli, Euler, and Gauss on what later got here to be often called Fourier sequence. J. Fourier in his 1822 Theorie analytique de los angeles Chaleur [16] (still on hand as a Dover reprint) was once the 1st to assert that arbitrary periodic services may be increased in a trigonometric (later known as a Fourier) sequence, a declare that was once finally proven to be mistaken, even though now not too faraway from the reality. it really is an a laugh old sidelight that this paintings gained a prize from the French Academy, even with critical matters expressed through the judges (Laplace, Lagrange, and Legendre) re­ garding Fourier's loss of rigor.

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O" ............................. ~..... ~. 34: Sampled Sinusoid: Sampling Period 1/31r simple example shows that discrete time signals obtained from continuous time signals can be quite different in appearance and behavior even if the original signal being sampled is fixed. , modify) shifted versions of a waveform. 1. This pulse is an example of a timelimited signal, an infinite duration signal that is nonzero only on an interval CHAPTER 1. 8. SAMPLING, WINDOWING, AND EXTENDING 41 of finite length.

For example, suppose that one observes a continuous time sine wave 36 CHAPTER 1. 31: Mona Lisa produced by a oscillator for T seconds. This could be modelled by a finite duration waveform {sin(wt); t E [0, Tn, but it might be useful to consider it as a piece of an infinite duration sine wave {sin(wt); tEn} or even as a time-limited waveform that lasts forever and assumes the value for t not in [0, T). Which model is more "correct"? None; the appropriate choice for a particular problem depends on convenience and the goal of the analysis.

21: The Wedge or Triangle Signal I\T(t) The operation of shifting or delaying a signal by T can be viewed as a system: the original signal put into the system produces an output that is a delayed version of the original signal. 6. 22: Shifted Triangle Signall\T(t - 2T) C defined by Ct(v) = v(t - r). It should be obvious that this system is a linear system. The idea of a shift extends in a natural fashion to finite duration signals by taking advantage of the relation between a finite duration signal and an infinite duration signal formed by replicating the finite duration signal forever: its periodic extension.