Fourier Transform as a Correlation

STUDENTS often contribute significant effort to gain intuition behind the Fourier Transform (and similarly Fourier Series). Recipies, smoothies, cakes, etc…

The mechanics of using integration to acheive the …

Fourier transform derivation from a correlation perspective… A cross-correlation integral is defined as

$$\begin{align} (f\star g)(\tau) &\triangleq \int_\infty^\infty \overline{f(t)}g(t+\tau)dt \newline &\triangleq \int_\infty^\infty \overline{f(t-\tau)}g(t)dt \end{align}$$

For discrete function, a cross-correlation sum is defined as

$$\begin{align} (f\star g)[n] &\triangleq \sum_\infty^\infty \overline{f[m]}g[m+n] \newline &\triangleq \sum_\infty^\infty \overline{f[m-n]}g[m] \end{align}$$

Let $f(t)$ be a bounded, differentiable function and $g(t)=e^{j\omega t}$ be a sinusoid.