Harmonically Related Complex Exponentials

Or actually, Harmonically related Signal building blocks.

Why?

Mental Model

Well, they usually are nothing more than that; They are signals, but not really. Most of the time when we study them, we only view them as building blocks for actual signals that we’re interested in; The same way the x and y components in a vector are vectors but not really. you usually don’t view them as actual vectors when their only usage to u is building other vectors, for example:

$$\underset{\text{actual vector}}{\underbrace{\left[\begin{array}{c}x\\ y\end{array}\right]}}=\underset{\text{some block}}{\underbrace{\left[\begin{array}{c}x\\ 0\end{array}\right]}}+\underset{\text{another block}}{\underbrace{\left[\begin{array}{c}0\\ y\end{array}\right]}}$$

Periodic Complex exponentials, when you only use them as building blocks for (say) a cosine signal, they get the same treatment (sad):

$$\underset{\text{actual signal}}{\underbrace{2\cdot \cos (\omega t)}}=\underset{\text{some block}}{\underbrace{e^{j\omega t}}}+\underset{\text{another block}}{\underbrace{e^{-j\omega t}}}$$

What a Periodic Complex Exponential is

Algebraic Form: $x(t)=C\cdot e^{j\omega t}$, where $\omega$ is its frequency and $C$ is a complex constant ($C=r{e}^{j\varphi }$)

$C$ stores both its length and starting/phase angle (u can think of it as its “initial position”):

• $r=|C|$ (radius)
• $\varphi =\angle C$ (phase angle)

Visual Form: A rotating vector that returns to its start after period $T_0=\frac{2\pi }{{\omega }_0}$.

In Signal Analysis, smart ppl noticed that if you combine random signal blocks, the result is usually a meaningless, chaotic wave that never repeats itself. But they noticed that some building blocks work together perfectly. When you stack these specific blocks together, the result is a clean signal that also repeats over and over (is periodic). ^argh They investigated what these “compatible” blocks had in common (cuz clean is good) and noticed a pattern: their frequencies ($\omega$) weren’t random; they were all integer multiples of a single “base” frequency (${\omega }_0$). In other words, they were harmonically related.

Now that they had a way to identify blocks that play nice together, they grouped them into families based on that shared ${\omega }_0$. In an ${\omega }_0$-family, the $k$-th member (harmonic) has a frequency of $\omega =k\cdot {\omega }_0$ and takes the form:

$${\varphi }_k(t)=e^{jk{\omega }_{0}t},k=0,\pm 1,\pm 2,...$$

Also, its fundamental period is $k$ times smaller than their shared period ($T_0$).

$$T_{ok}=\frac{T_0}{|k|}$$

This diagram shows how a family of Periodic Complex exponentials with some ($w_0$) looks like.

Visual

The tool below shows how different harmonics “build” up a signal $x(t$) over time $t$ (the time axis extends down against the real values of the signal, and to the right against the imaginary values of the signal).

The visualization shows a complete period of x(t), and since its fundamental period $T_0$ was set to $2\pi$ seconds, it runs for that long

You can use this tool to actually “see” what a signal that you’ve been solving a problem for looks like =)

Have fun.

Attribution

That tool was built by Kazad (over BetterExplained).

I simply took the source code and shamelessly vibe-coded it to make the visualization more intuitive to me.

Formal Definition

• a general complex exponential signal x(t)
• the kth harmonic of fund freq $\omega$

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Connections