Numerical Methods Continuous Fourier Series Part: Continuous Fourier Series

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Chapter 11.02: Continuous Fourier Series Lecture # 2 For a function with period “T” continuous Fourier series can be expressed as The “average” function value between the time interval [0,T] is given by (22) (23)

Even and Odd functions are described as (24) (25) Continuous Fourier Series

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

(27) (28) Derivation of formulas for

Now, if both sides are multiplied by and then integrated (29) 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) Derivation of formulas for

Similar derivation can be used to obtain Derivation of formulas for

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Numerical Methods Continuous Fourier Series Part: Example 1

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Chapter 11.02: Example 1 (Contd.) Lecture # 3 Using the continuous Fourier series to approximate the Find the Fourier coefficients and Figure 1: A Periodic Function following periodic function in Fig (1).

Example 1 cont. From Equations (23-25), one obtains :

Example 1 cont.

From this equation, we obtain the Fourier coefficients for = = = = = = = = Example 1 cont.

We can now find the values of Example 1 cont. from the following equations,

For the Fourier coefficients can be = = e-6  0 = = e-6  0 = = e-6  0 = = e-6  0 Example 1 cont. computed as

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

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Numerical Methods Continuous Fourier Series Part: Complex Form of Fourier Series

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Chapter :Complex form of Fourier Series (Contd.) Lecture # 4 Using Euler’s identity (31) (32) and

Thus, the Fourier series can be casted in the following form: (33) (34) Complex form of Fourier Series cont.

Define the following constants: Hence: (35) (36) (37) Complex form of Fourier Series cont.

Using the even, odd (38) properties Equation (37) becomes: Complex form of Fourier Series cont.

Complex form of Fourier Series cont. Substituting Equations (35,36, 38) into Equation (34), one gets:

or (39) Complex form of Fourier Series cont.

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

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

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