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. Two examples are shown below.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
Since the multiplicative factors of in the periods for cos t are whole numbers but the corresponding factors
for cos(p2 t) are irrational (always involving p2) there is no number that appears in both lists. Hence the
function f(t) = cost + cos(p2t) is NOT periodic (it never repeats itself), despite its comparatively simple
form.
More generally, we can say that the sum of two or more cosine waves will be periodic only when the ratios
of all pairs of periods form rational numbers (ratios of integers).
Sine wave Let us recall de nitions linked to a sine wave
................................................................................................................................................................................................... .........
.......................................................................
........
f
0 t
f0 sin
? =! 2=!
...................................................................................................................................................................Figure 2
.... ..... ..... ..... ..... ..... ..... ...... ...... ...... ..... ..... ..... .....
......
.......
.......
.......
.......
.......
....
...
...
Consider f(t) = f0 sin(!t + ) = f0 sin(2t + ), shown above, where
f0 is amplitude,
! is the circular (or angular) frequency in radians/unit time,
is frequency in cycles/unit time (Hertz )
is phase angle with respect to time origin in radians.
The period of above sine wave is 1= = 2=! seconds.
A positive phase angle shifts waveform to the left (a lead) and a negative phase angle moves waveform to
the right (a lag).
2. Whole range Fourier series
Fourier analysis decomposes a `complicated periodic wave shape into a sum of sine and cosine waves.
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 .
|