engineering and numerical analyses 6.pdf
- 1. Periodic Functions and Fourier Series University of Technology Materials Engineering Dept. Numerical and Engineering Analysis By Dr. Kadhum Muttar
- 2. • What is Fourier Series? – Representation of a periodic function with a weighted, infinite sum of sinusoids. • Why Fourier Series? – Any arbitrary periodic signal, can be approximated by using some of the computed weights – These weights are generally easier to manipulate and analyze than the original signal • What is a periodic Function? – A function which remains unchanged when time-shifted by one period f(t) = f(t + T) or f(t) = f(t + 2p) Where T is the period of periodic function (T = 2p)
- 3. 0 θ T f θ ( ) 0 θ T f θ ( )
- 5. Fourier Series ( ) t f If is a periodic function with period p 2 The function can be represented by a trigonometric series as: ( ) ) 1 ( sin cos 2 1 1 0 ∑ ∑ ∞ = ∞ = + + = n n n n p t n b p t n a a t f π π
- 6. We want to determine the coefficients, n a and n b . Let us first remember some useful integrations. 0 = θ θ θ ∫ π π − d m n cos cos m n ≠ π = θ θ θ ∫ π π − d m n cos cos m n =
- 7. 0 = θ θ θ ∫ π π − d m n cos sin for all values of m. 0 = θ θ θ ∫ π π − d m n sin sin m n ≠ π = θ θ θ ∫ π π − d m n sin sin m n =
- 8. Determination of a0 Integrate both sides of Eq. (1) ( ) dt p t n b p t n a a dt t f p d d n n n n p d d ∫ ∑ ∑ ∫ + ∞ = ∞ = + + + = 2 1 1 0 2 sin cos 2 π π ( ) 0 0 2 1 2 0 2 + + = ∫ ∫ + + dt a dt t f p d d p d d ( )dt t f p a p d d ∫ + = 2 0 1
- 9. Determine n a Multiply Eq.(1) by p t nπ cos and then integrate both sides from d to d+2p ( ) dt p t n p t n b p t n a a dt p t n t f p d d n n n n p d d ∫ ∑ ∑ ∫ + ∞ = ∞ = + + + = 2 1 1 0 2 cos sin cos 2 cos π π π π ( ) dt p t n t f p a p d d n π cos 1 2 ∫ + =
- 10. Determine n b Multiply Eq.(1) by p t nπ sin and then Integrate both sides from d to d+2p ( ) dt p t n p t n b p t n a a dt p t n t f p d d n n n n p d d ∫ ∑ ∑ ∫ + ∞ = ∞ = + + + = 2 1 1 0 2 sin sin cos 2 sin π π π π ( ) dt p t n t f p b p d d n π sin 1 2 ∫ + =