Fourier analysis keeps our attention focused on frequency content, the $\omega_0$ in $e^{i\omega_0 n}$. Note that $\omega_0 = \frac{w\pi}{p}$, where $p$ is the \emph{period}, and $\omega_0$ is the \emph{fundamental frequency}. Every signal can be decomposed into linear combinations of harmonically related complex exponentials. Harmonically related simply means each has a frequency that is an integer multiple of the fundamental frequency. Assume for now that this decomposition can be done. Then we can sum over any continguous interval of length $p$ and a signal can be written as

$$ x(n) = \sum \limits_{k=0}^{p-1}X_ke^{i k \omega_0 n} $$

Lets compare this with the decomposition of a signal as shifted impulses:

$$ x(n) = \sum \limits_{m\in\Z} x(m) \delta(n-m)  $$

When we pass this signal through a system $h$, we get the convolution sum. Lets derive this through the properties of a linear time invariant system $h$:
\begin{align*}
\delta(n) \to &\fbox{h} \to h(n) \\
\delta(n-m) \to &\fbox{h} \to h(n-m) \\
x(m)\delta(n-m) \to &\fbox{h} \to x(m)h(n-m) \\
\sum_m (m)\delta(n-m) \to &\fbox{h} \to \sum_m x(m)h(n-m) \\
\end{align*}

We know what happens when we pass a complex exponential into an LTI system:
\begin{align*}
e^{i\omega_0n} \to & \fbox{H} \to H(\omega_0)e^{i\omega_0n} \\
e^{ik\omega_0 n}\to & \fbox{H} \to H(k\omega_0)e^{ik\omega_0 n} \\
X_ke^{ik\omega_0 n}\to & \fbox{H} \to X_kH(k\omega_0)e^{ik\omega_0 n} \\
\sum_k e^{ik\omega_0 n}\to & \fbox{H} \to \sum_k X_kH(k\omega_0)e^{ik\omega_0 n} \\
\end{align*}
So we can generalize this for a decomposed signal $x(n) = \sum_{k=0}^{p-1}X_ke^{i k \omega_0 n}$:
$$ x(n) = \sum \limits_{k=0}^{p-1}X_ke^{i k \omega_0 n} \to \fbox{H} \to \sum \limits_{k=0}^{p-1}X_kH(k \omega_0)e^{i k \omega_0 n}  $$

\begin{claim}
Given a discrete fourier series expansion of a signal $x(n) = \sum_{k=0}^{p-1}X_ke^{i k \omega_0 n}$, the signal convolved with a signal $h$ can be given by:
$$(x*h)(n) =  \sum \limits_{k=0}^{p-1}X_kH(k \omega_0)e^{i k \omega_0 n}  $$
\end{claim}

\begin{proof}
\begin{align*}
(x*h)(n) &= \sum \limits_{k=-\infty}^{\infty} h(k) x(n-k) \\
(x*h)(n) &= \sum \limits_{k=-\infty}^{\infty} h(k) \sum_{\ell=0}^{p-1}X_\ell e^{i \ell \omega_0 (n-k)} \\
(x*h)(n) &= \sum \limits_{k=-\infty}^{\infty} h(k) \sum_{\ell=0}^{p-1}X_\ell e^{i \ell \omega_0 n}e^{-i \ell \omega_0k} \\
(x*h)(n) &=  \sum_{\ell=0}^{p-1} \sum \limits_{k=-\infty}^{\infty} h(k) X_\ell e^{i \ell \omega_0 n}e^{-i \ell \omega_0k} \\
(x*h)(n) &=  \sum_{\ell=0}^{p-1} X_\ell \lr{\sum \limits_{k=-\infty}^{\infty} h(k) e^{-i \ell \omega_0k}} e^{i \ell \omega_0 n}\\
(x*h)(n) &=  \sum_{\ell=0}^{p-1} X_\ell H(\ell\omega_0) e^{i \ell \omega_0 n}\\
\end{align*}
\end{proof}


Hence, we have the description of what happens to a signal when passed through a system. This mathematically describes what an audio equalizer does when we pass in a signal described by its constituent frequencies.


Harmonic Relations and Frequency Content