Periodic signals
Fourier series are used in many areas of engineering, and most of you will discuss the method again in your
second year mathematics units. We consider here Fourier series expansions of periodic functions, i.e.
functions which repeat themselves exactly at regular intervals.
Def. A function f is periodic of period T (T > 0) if and only if f(t + T) = f(t) for all t.
Therefore the period T is de ned as the time interval required for one complete fluctuation.
Hence f(t) = cost is periodic with period 2 since
f(t + 2) = cos(t + 2) = cost = f(t) for all t:
N.B. If f is periodic with period T , then clearly from the graphs, or from repeated use of the de nition, f
is also periodic with periods 2T; 3T; : : : { you should choose the minimum period of the function to be its
period.
When the French mathematician Joseph Fourier (1768–1830) was trying to solve a problem
in heat conduction, he needed to express a function as an infinite series of sine and
cosine functions:
Earlier, Daniel Bernoulli and Leonard Euler had used such series while investigating problems
concerning vibrating strings and astronomy.
The series in Equation 1 is called a trigonometric series or Fourier series and it turns
out that expressing a function as a Fourier series is sometimes more advantageous than
expanding it as a power series. In particular, astronomical phenomena are usually periodic,
as are heartbeats, tides, and vibrating strings, so it makes sense to express them in terms
of periodic functions.
المادة المعروضة اعلاه هي مدخل الى المحاضرة المرفوعة بواسطة استاذ(ة) المادة . وقد تبدو لك غير متكاملة . حيث يضع استاذ المادة في بعض الاحيان فقط الجزء الاول من المحاضرة من اجل الاطلاع على ما ستقوم بتحميله لاحقا . في نظام التعليم الالكتروني نوفر هذه الخدمة لكي نبقيك على اطلاع حول محتوى الملف الذي ستقوم بتحميله .
الرجوع الى لوحة التحكم
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