Numerical Methods Continuous Fourier Series Part Continuous Fourier

Numerical Methods Continuous Fourier Series Part: Continuous Fourier Series http: //numericalmethods. eng. usf. edu

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Lecture # 2 Chapter 11. 02: Continuous Fourier Series For a function with period “T” continuous Fourier series can be expressed as (22) The “average” function value between the time interval [0, T] is given by (23) 5 lmethods. eng. usf. edu ht

Continuous Fourier Series Even and Odd functions are described as (24) (25) 6 lmethods. eng. usf. edu ht

Derivation of formulas for Integrating both sides with respect to time, one gets (26) The second and third terms on the right hand side of the above equations are both zeros 7 lmethods. eng. usf. edu ht

Derivation of formulas for (27) (28) 8 lmethods. eng. usf. edu ht

Derivation of formulas for Now, if both sides are multiplied by and then integrated (29) 9 lmethods. eng. usf. edu ht

Derivation of formulas for The first and second terms on the RHS of Equation (29) are zero. The third RHS term of Equation (29) is also zero, with the exception when (30) 10 lmethods. eng. usf. edu ht

Derivation of formulas for Similar derivation can be used to obtain 11 lmethods. eng. usf. edu ht

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Numerical Methods Continuous Fourier Series Part: Example 1 http: //numericalmethods. eng. usf. edu

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Lecture # 3 Chapter 11. 02: Example 1 (Contd. ) Using the continuous Fourier series to approximate the following periodic function Find the Fourier coefficients in Fig (1). and Figure 1: A Periodic Function 20 lmethods. eng. usf. edu ht

Example 1 cont. From Equations (23 -25), one obtains : 21 lmethods. eng. usf. edu ht

Example 1 cont. 22 lmethods. eng. usf. edu ht

Example 1 cont. From this equation, we obtain the Fourier coefficients for = = = = 23 -0. 9999986528958207 -0. 4999993232285269 -0. 3333314439509194 -0. 24999804122384547 -0. 19999713794872364 -0. 1666635603759553 -0. 14285324664625462 -0. 12499577981019251 lmethods. eng. usf. edu ht

Example 1 cont. We can now find the values of equations, 24 from the following lmethods. eng. usf. edu ht

Example 1 cont. For computed as the Fourier coefficients = = = = 25 -0. 6366257003116296 -5. 070352857678721 e-6 -0. 07074100153210318 -5. 070320092569666 e-6 -0. 025470225589332522 -5. 070265333302604 e-6 -0. 012997664818977102 -5. 070188612604695 e-6 can be 0 0 lmethods. eng. usf. edu ht

Example 1 conclusion In conclusion, the periodic function f(t) (shown in Figure 1) can be expressed as: where 26 For and have already computed one obtains: lmethods. eng. usf. edu ht

27 lmethods. eng. usf. edu ht

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Acknowledgement This instructional power point brought to you by Numerical Methods for STEM undergraduate http: //numericalmethods. eng. usf. edu Committed to bringing numerical methods to the undergraduate

For instructional videos on other topics, go to http: //numericalmethods. eng. usf. edu/videos/ This material is based upon work supported by the National Science Foundation under Grant # 0717624. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.

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Numerical Methods Continuous Fourier Series Part: Complex Form of Fourier Series http: //numericalmethods. eng. usf. edu

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Lecture # 4 Chapter 11. 02 : Complex form of Fourier Series (Contd. ) Using Euler’s identity and (31) 36 lmethods. eng. usf. edu (32)ht

Complex form of Fourier Series cont. Thus, the Fourier series can be casted in the following form: (33) (34) 37 lmethods. eng. usf. edu ht

Complex form of Fourier Series cont. Define the following constants: (35) (36) Hence: (37) 38 lmethods. eng. usf. edu ht

Complex form of Fourier Series cont. Using the even , odd properties Equation (37) becomes: (38) 39 lmethods. eng. usf. edu ht

Complex form of Fourier Series cont. Substituting Equations (35, 36, 38) into Equation (34), one gets: 40 lmethods. eng. usf. edu ht

Complex form of Fourier Series cont. or (39) 41 lmethods. eng. usf. edu ht

Complex form of Fourier Series cont. The coefficient can be computed, by substituting Equations (24, 25) into Equation (36) to obtain: (40) or 42 lmethods. eng. usf. edu ht

Complex form of Fourier Series cont. Substituting Equations (31, 32) into the above equation, one gets: (41) Thus, Equations (39, 41) are the equivalent complex version of Equations (21, 25). lmethods. eng. usf. edu 43 ht

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Acknowledgement This instructional power point brought to you by Numerical Methods for STEM undergraduate http: //numericalmethods. eng. usf. edu Committed to bringing numerical methods to the undergraduate

For instructional videos on other topics, go to http: //numericalmethods. eng. usf. edu/videos/ This material is based upon work supported by the National Science Foundation under Grant # 0717624. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.

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