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465: Examining Fourier Synthesis with Dirac Notation

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  • The purpose of this tutorial is to use Dirac notation to examine Fourier synthesis. The first step is to write the function symbolically in Dirac notation.

    \[f(x)=\big \langle x|f\big \rangle\]

    Select an orthonormal basis set, |n>, for which the completeness relation holds

    \[\sum_{n}|n \big \rangle \big \langle n|=1\]

    Expand |f> in terms of |n> by inserting equation (2) into the right side of equation (1). In other words write f(x) as a weighted () superposition using the basis set (the |n> basis set expressed in the coordinate representation).

    \[f(x)=\sum_{n} \big \langle x|n\big \rangle \big \langle n|f\big \rangle\]

    Evaluate the Fourier coefficient, , using the continuous completeness relation in coordinate space.

    \[\int |x' \big \rangle\big \langle x'|dx'=1\]

    Equation (3) becomes,

    \[f(x)=\sum_{n} \big \langle x|n \big \rangle \int \big \langle n|x' \big \rangle\big \langle x'|f \big \rangle dx'\]

    Now select a function

    \[\big \langle x'|f \big \rangle = x'^{3}(1-x')\]

    over the interval (0,1). Choose the following orthonormal basis set over the same interval.

    \[\big \langle x|n \big \rangle = \sqrt{2}sin(n \pi x)\]

    Substitution of equations (6) and (7) into (5) yields

    \[f(x) = \sum_{n} \sqrt{2}sin(n \pi x) \int_{0}^{1} \sqrt{2}(n \pi x')x'^{3}(1-x')dx'\]

    The Fourier synthesis and the original function are shown for n = 2, 4, and 10 in the figure below.

    \(x:=0,.025 ..1.0\)

    \(f(x,n) :=\sum_{i=1}^{n}[ \sqrt{2}\cdot sin(i \cdot \pi \cdot x) \cdot \int_{0}^{1} \sqrt{2} \cdot sin(i \cdot \pi \cdot x') \cdot x'^{3} \cdot (1-x')dx')]\)

    Screen Shot 2019-01-29 at 8.26.46 PM.png