Frequency spectrum

In Fourier series analysis, the frequency spectrum of a signal $x$ is the set:

$$\{{\alpha }_k{\}}_{k=-K}^K$$

i.e., the Fourier series coefficients of $x$, with $k\in \mathbb{Z}$. This represents $x(t)$ in the frequency domain, which consists of multiples of the fundamental frequency. These are easy representations of the signal!

Take this example,¹ where:

$$x(t)=\cos (4\pi t)+2\sin (6\pi t)$$

From the orthogonality property, we have four frequencies with non-zero coefficients:

Diagram construction

Footnotes

1. From Prof Simpson-Porco’s lecture slides. ↩