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Fourier transform

The Fourier transform is a technique that is used to transform a function to a series of sinusoidal functions. It can be derived from a special case of the Fourier series as period $T\to\infty$. The Fourier series is expressed as \begin{equation} f(t)=\sum_{n=-\infty}^\infty A_ne^{2i\pi nt/T} \label{eq:fourier_series} \end{equation} where \begin{equation} A_n=\frac{1}{T}\int_0^T f(t)e^{-2i\pi nt/T}\,dt. \label{eq:A_n} \end{equation}

We can rewrite Eq. \eqref{eq:A_n} as \begin{equation} TA_n=\int_0^T f(t)e^{-2i\pi nt/T}\,dt \label{eq:TA_n} \end{equation} and Eq. \eqref{eq:fourier_series} as \begin{equation} f(t)=\sum_{n=-\infty}^\infty TA_ne^{2i\pi nt/T}\frac{1}{T} \label{eq:fourier_series_2} \end{equation}

“Intuitively,” as $T\to\infty$, it expands into both positive and negative $t$-directions, so the range of integral changes from $[0,T]$ to $(-\infty,\infty)$ in Eq. \eqref{eq:TA_n}. We can define a new variable $u=\frac{n}{T}$ which can take on any value because $n\in(-\infty,\infty)$ and $\frac{1}{T}\to 0$. Rewrite Eq. \eqref{eq:TA_n} to define the forward Fourier transform (analysis equation) \begin{equation} F(u)=\int_{-\infty}^\infty f(t)e^{-2i\pi ut}\,dt. \label{eq:analysis} \end{equation} Again, as $T\to\infty$, $\frac{1}{T}\to 0$ and we can write $\frac{1}{T}$ as $du$. Now, rewrite Eq. \eqref{eq:fourier_series_2} as an integral to derive the inverse Fourier transform (synthesis equation) \begin{equation} f(t)=\int_{-\infty}^\infty F(u)e^{2i\pi ut}\,du. \label{eq:synthesis} \end{equation}