Beregn Fourier-transformen av rektangelpulsen f(x)=1f(x)=1f(x)=1 for ∣x∣<a|x|<a∣x∣<a, 000 ellers.
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f^(w)=12π∫−aae−iwxdx=12π[e−iwx−iw]−aa=12π⋅2sin(aw)w=2π sin(aw)w\displaystyle \hat f(w)=\frac{1}{\sqrt{2\pi}}\int_{-a}^{a}e^{-iwx}dx=\frac{1}{\sqrt{2\pi}}\left[\frac{e^{-iwx}}{-iw}\right]_{-a}^{a}=\frac{1}{\sqrt{2\pi}}\cdot\frac{2\sin(aw)}{w}=\sqrt{\frac{2}{\pi}}\,\frac{\sin(aw)}{w}f^(w)=2π1∫−aae−iwxdx=2π1[−iwe−iwx]−aa=2π1⋅w2sin(aw)=π2wsin(aw). Dette er en sinc-funksjon.
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