Physics 434, 2012: Lecture 17

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Back to Physics 434, 2012: Information Processing in Biology. Here we introduce the idea of Fourier series and Fourier transforms. We have discussed in the previous lecture why we need them.

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1. Consider a function $f(t)$ periodic on $-\pi \leq t\leq \pi$.
2. We would like to approximate this function as $f(t)=\sum _{k=0}^{N}a_{k}\cos kt+\sum _{k=1}^{N}b_{k}\sin kt$, takin $N\to \infty$ at some point.
3. From this expression, we can find the coefficients Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle a_{k},b_{k}} self-consistently. Indeed, let's multiply the equation by Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \cos mt} and integrate from Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle -\pi } to $\pi$.
4. All terms containing products Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \cos kt\sin mt} are zero.
5. For the $\cos kt\cos mt$ terms, we have Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \cos kt\cos mt=1/2(\cos(k+m)t+\cos(k-m)t)} .
6. Completing the integrals, we have: Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \int _{-\pi }^{\pi }dtf(t)\cos mt=\sum _{m}\int _{-\pi }^{\pi }dt{\frac {a_{k}}{2}}\left(\cos(k+m)t+\cos(k-m)t\right)=\sum _{m}\pi a_{k}(\delta _{k,m}+\delta _{k,0}\delta _{m,0})=\pi a_{m}(1+\delta _{m,0}} .
7. We can do similar to find Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle b_{k}} : multiply by Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \cos mt} , and integrate.
8. This gives: Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle a_{k>0}={\frac {1}{\pi }}\int _{-\pi }^{\pi }dtf(t)\cos k_{t}} , Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle a_{0}={\frac {1}{2\pi }}\int _{-\pi }^{\pi }dtf(t)} , Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle b_{k>0}={\frac {1}{\pi }}\int _{-\pi }^{\pi }dtf(t)\sin k_{t}} .