SU-ENGR76 APR182024 — Jemoka Knowledge Base

Fourier Series as exactly a shifted sum of sinusoids Key idea: every periodic function with period L can be represented as a sum of sinusoids

\begin{equation} f(t) = A_0 + \sum_{i=1}^{\infty} B_{j} \sin \left(k \omega t + \phi_{j}\right) \end{equation}

where \omega = \frac{2\pi}{T}. notice! without the A_0 shift, our thing would integrate to 0 for every L; hence, to bias the mean, we change A_0. Now, we ideally really want to get rid of that shift term \phi, applying the sin sum formula:

\begin{align} f(t) &= A_0 + \sum_{i=1}^{\infty} B_{j} \sin \left(k_{j} \omega t + \phi_{j}\right) \\ &= A_0 + \sum_{j=1}^{\infty } A_{j} \cos \left(\phi_{j}\right) \sin \left(k_{j}\omega t\right) + B_{j} \sin \left(\phi_{j}\right) \cos \left(k_{j} \omega t\right) \\ &= b_0 + \sum_{j=1}^{\infty} a_{j} \sin \left(\omega k_{j} t\right) + \sum_{j=1}^{\infty} b_{j} \cos \left(k \omega t\right) \end{align}

we can move back and fourth before the representation as follows:

\begin{equation} \begin{cases} a_{j} = A_{j} \cos \left(\phi_{j}\right) \\ b_{j} = A_{j} \sin \left(\phi_{j}\right) \\ b_{0} = A_{0} \\ A_{j}^{2} = a_{j}^{2} + b_{j}^{2} \\ \tan \left(\phi_{j}\right) = \frac{b_{j}}{a_{j}} \end{cases} \end{equation}

in a sense, this is a polar representation of the sum of sinusoids of system. Recall to get the actual coefficients, see General Fourier Decomposition. signal representation KEY IDEA: we can approximate all values of a function with just specifying the parameters of the sine function. In particular, any signal f is uniquely specified by specifying only its Fourier representation:

\begin{equation} \left\{A_{j}, \phi_{j}\right\}_{0}^{\infty} \cup \{A_0\} \end{equation}

The smallest f_{1} = \frac{1}{T}, called the fundamental frequency of this system, and any higher are harmonics; in particular, f_{j} = \frac{j}{T} are called the jth harmonic. To represent finite-duration signal, we just create the finite-periodic extension of this signal by coping it over and over. Key thing to remember: remember that odd/even extensions have period 2T!!!