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Fourier Series

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  Disediakan oleh : Pn Misida Binti Senon FOURIER SERIES  A periodic function f(t) can be represented by an infinite sum of sine and/or cosine functions that are harmonically related. That is, the frequency of any trigonometric term in the infinite series is an integral multiple, or harmonic, of the fundamental frequency of the periodic function. Thus, given f(t) is periodic (e.g. square wave, triangular wave, half rectified wave, etc.), then f(t) can be represented as follows: ----- (Eq. 1.1) where n is the integer sequence 1,2,3, ... , a 0 , a n , and b n  are called the Fourier coefficients, and are calculated from f(t), 0    = T     2  is the fundamental frequency of the periodic function f(t) with period T, and 0   n is known as the n-th harmonic of f(t). Periodic Functions . In terms of Eq. (1.1), f(t) can be extended to   t  ; it becomes a periodic function, with a period 2 π   , i.e. , f(t) = f(t + 2 π ). For instance, instead of [0, 2 π ], we can use [ −π, π ] for limits in Eq. (I.2). dt t   f  T a T T   )(1 220       ------- dt t   f  a  )(21 0               ------ dt t   f  a  )(21  200             tdt nt   f   T a T T n    cos)(1 22     ---- tdt nt   f  a n            cos)(21     ----  tdt nt   f  a n         cos)(21  20     tdt nt   f   T b T T n    sin)(1 22     ---- tdt nt   f  b n            sin)(21     ----  tdt nt   f  b n         sin)(21  20      Disediakan oleh : Pn Misida Binti Senon Example 1 : Show that f(x) = 5 sin 3 x is a periodic function of period 2π.   Solution  : f (x + 2 π ) = 5 [ (  )]  = 5 [ ]   where cos 2π = 1 ; sin 2π = 0  =  []  = 5 sin 5x = f (x) ∴ f (x + 2 π) = f(x) Shown   [()]   The effect of symmetry on the Fourier coefficients Even function symmetry  A function is defined as even if f(t) = f(-t). The equations for the Fourier coefficients reduce to:    Example : t t   f    cos)(     )'()'(  t   f  t   f        f(t) - t’ t’   t  f(t)  Disediakan oleh : Pn Misida Binti Senon Odd function symmetry  A function is defined as odd if f(t) = -f(-t); then the Fourier coefficients are given by:    Example ; t t   f    sin)(     i. ; ii. ; iii. ; for all n    f(t) t  f(t) t  f(t) t i. ; ; for all n,  ii. ; ; for all n,  iii. ;  Disediakan oleh : Pn Misida Binti Senon EXAMPLE 2 : If f(x) =      is defined in the interval  –    π < x < π and f(x) = f (x + 2π);  a) state whether the function is odd or even. b) determine which components of the Fourier Series are present in the function. EXAMPLE 3: Sketch the waveform and label the axis, period and the amplitude of the periodic function defined as :  30)( t   f     ,,   5005  t t    T = 10 a o dx x  f  T       55 )(1       23151013101301 000          xdxdxT           f(x) -10 -5 0 5 10 15 T 3 T=10

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Jul 23, 2017

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