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Chapter 3 The Fourier Series EE 207 Adil S. Balghonaim

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Vectors in 2D Vectors in 3D N-Dimension

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Jean Baptiste Joseph Fourier (21 March 1768 – 16 May 1830) was a French mathematician and physicist best known for initiating the investigation of Fourier series and their applications to problems of heat transfer and vibrations

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Fourier observed that the addition of Sinusoidal functions with different frequencies and amplitude resulted in other periodical functions Example

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Fourier proposed the following: For any periodical function of period were Then Periodicals functions can be decomposed to sin and cosine functions Similar to decompose a vector in terms of i,j,k,…

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We seek to find the coefficients or coordinates

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Integrating both side over one period

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By comparison to the 2D-vector, Similarly

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Multiplying both side by cos5 0 t and Integrating over one period

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Integrating sinusoidal over one period

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The average of x(t)

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Example 3-4 The average value of x(t) = 0

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Now for n =`1

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term 1 and term 2 are complex conjugate of each other Then we can write x(t) as Where

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Since

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Since it is true for all m then it is true for all n

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Example 3-6 Find the complex Fourier series coefficients for A half-rectified sine wave

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Similarly

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First Entry in Table 3-1

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Table 3-1(old Book)

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Symmetry Properties of Fourier Series coefficients

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Line Spectra where In general a complex number that can be represented as a phasor Is a rotating phasor of frequency Therefore, x(t) consists of a summation of rotating phasors

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ME 440 Intermediate Vibrations Tu, January 27, 2009 Sections 1.10 & 1.11 © Dan Negrut, 2009 ME440, UW-Madison.

ME 440 Intermediate Vibrations Tu, January 27, 2009 Sections 1.10 & 1.11 © Dan Negrut, 2009 ME440, UW-Madison.

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