6.3.2.4 Fourier transform ¶

In Fourier series, we had to assume that the function is periodic outside of the desired interval with a period of $L$. Therefore, assuming that $L\rightarrow\infty$ will allow us to work with any function. However, with this approximation, the fundamental frequency ($\omega_0$) or the frequency resolution that we discussed in Fourier series will tend to zero: $\omega_0\rightarrow0$. In the equation to find $c_m$, every $m$ represented a frequency (multiple of $\omega_0$) and the integration on $l$ removes the dependence of the right side of the equation on $l$, making it only a function of $m$ or frequency. Let’s define the following two variables:

$$\omega{\equiv}m\omega_0={2{\pi}m\over L}$$

$$F(\omega){\equiv}Lc_m$$

The equation to find the coefficients of each frequency in Fourier series thus becomes:

$$F(\omega)=\int_{-\infty}^{\infty}f(l)e^{-i{\omega}l}dl.$$

The function $F(\omega)$ is thus the Fourier transform of $f(l)$ in the frequency domain. So through this transformation, we can find (analyze) the magnitudes of the constituting frequencies or the value in the frequency space¹⁷³ of our spatial input function. The great thing is that we can also do the reverse and later synthesize the input function from its Fourier transform. Let’s do it: with the approximations above, multiply the right side of the definition of the Fourier Series (Fourier series) with $1=L/L=({\omega_0}L)/(2\pi)$:

$$f(l)={1\over 2\pi}\displaystyle\sum_{n=-\infty}^{\infty}Lc_ne^{{2{\pi}in\over L}l}\omega_0={1\over 2\pi}\displaystyle\sum_{n=-\infty}^{\infty}F(\omega)e^{i{\omega}l}\Delta\omega$$

To find the right most side of this equation, we renamed $\omega_0$ as $\Delta\omega$ because it was our resolution, $2{\pi}n/L$ was written as $\omega$ and finally, $Lc_n$ was written as $F(\omega)$ as we defined above. Now, as $L\rightarrow\infty$, $\Delta\omega\rightarrow0$ so we can write:

$$f(l)={1\over 2\pi}\int_{-\infty}^{\infty}F(\omega)e^{i{\omega}l}d\omega$$

Together, these two equations provide us with a very powerful set of tools that we can use to process (analyze) and recreate (synthesize) the input signal. Through the first equation, we can break up our input function into its constituent frequencies and analyze it, hence it is also known as analysis. Using the second equation, we can synthesize or make the input function from the known frequencies and their magnitudes. Thus it is known as synthesis. Here, we symbolize the Fourier transform (analysis) and its inverse (synthesis) of a function $f(l)$ and its Fourier Transform $F(\omega)$ as ${\cal F}[f]$ and ${\cal F}^{-1}[F]$.