Signals and systems/GF Fourier: Difference between revisions
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==Fourier series== |
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The Fourier series is used to analyze arbitrary periodic functions by showing them as a composite of sines and cosines. |
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A function is considered periodic if <math> x(t) = x(t+T)\, </math> for <math> T \neq 0 </math>. |
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The exponential form of the Fourier series is defined as <math> x(t) = \sum_{n=-\infty}^\infty \alpha_n e^{{j2\pi nt}/T} \, </math> |
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==Determining the coefficient <math> \alpha_n \,</math> == |
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<math> x(t) = \sum_{n=-\infty}^\infty \alpha_n e^{{j2\pi nt}/T} \, </math> The definition of the Fourier series |
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<math> \int_{-T/2}^{T/2} x(t)\, dt = \sum_{n=-\infty}^\infty \alpha_n \int_{-T/2}^{T/2} e^{{j2\pi nt}/T} dt</math> Integrating both sides for one period. The range of integration is arbitrary, but using <math> \int_{-T/2}^{T/2} </math> scales nicely when extending the Fourier series to a non-periodic function |
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<math> \int_{-T/2}^{T/2} x(t) e^{{-j2\pi mt}/T} dt = \sum_{n=-\infty}^\infty \alpha_n \int_{-T/2}^{T/2} e^{{j2\pi nt}/T}e^{{-j2\pi mt}/T} dt = \sum_{n=-\infty}^\infty \alpha_n \int_{-T/2}^{T/2} e^{{j2\pi (n-m)t}/T} dt</math> Multiply by the complex conjugate |
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<math> \int_{-T/2}^{T/2} x(t) e^{{-j2\pi mt}/T} dt = \sum_{n=-\infty}^\infty \alpha_n \frac{Te^{{j2\pi (n-m)t}/T}}{{j2\pi (n-m)}}</math> |
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== <math> \left \langle Bra \mid Ket \right \rangle </math> Notation == |
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==Linear Time Invariant Systems== |
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==Changing Basis Functions== |
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==Identities== |
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<math>e^{j \theta} = \cos \theta + j \sin \theta \, </math> |
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<math>\sin x = \frac{e^{jx}-e^{-jx}}{2j} \,</math> |
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<math>\cos x = \frac{e^{jx}+e^{-jx}}{2} \,</math> |
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<math> \left \langle n \mid m \right \rangle = T \delta_{n,m} \,</math> |
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