# Periodicity and waves

## Contents

# 1.2. Periodicity and waves#

This section introduces the concept of *periodicity*, which characterizes repeating signals. This idea is central to many concepts in both signal processing and perception of audio, so it’s worth spending some time with.

Specifically, this section covers the following topics:

Periodicity

Fundamental frequency

Waves (sinusoids) and their parameters

Basic properties of waves

## 1.2.1. Periodicity#

(Periodicity)

A signal \(x(t)\) is said to be *periodic* if there exists some finite \(t_0 > 0\) such that for every time \(t\),

The smallest such \(t_0\) satisfying this equation, if it exists, it is called the *fundamental period* (or sometimes just *period*) of the signal \(x\).
Different signals may have different periods, and some signals may have no period at all.

Think of the period of a signal as the shortest amount of time it takes, when listening to a looped recording, before you hear the recording start over.

Example (Pulse train)

A *pulse* train is a special kind of signal that consists only of ones and zeros, with the ones being separated at regular intervals.
An example pulse train with a spacing of 1.5 second between pulses can be defined mathematically as

and the first few seconds are visualized as Fig. 1.4.

Fig. 1.4 demonstrates the pulse train which has a repeating pulse every 1.5 seconds.
The signal is periodic with \(t_0 = 1.5\) because every point on the blue curve is identical to the point exactly \(t_0\) seconds later.
Note that this must hold for **every** time \(t\), not just the locations of the pulses!
The arrows indicate two such repetitions, though in general there will be infinitely many.

This points to a key consequence of periodicity.

Tip

If a signal is periodic, it must repeat forever:

This is why we take \(t_0\) to be the smallest possible period.

## 1.2.2. Aperiodic signals#

If a signal does not have a period, we call it *aperiodic*, and assign period \(t_0 = \infty\).
Think of this as meaning that you would have to wait (literally) forever before you see the signal repeat itself exactly.

Aperiodic signals come in all kinds of flavors. Random noise (e.g., white noise) is aperiodic, and if it wasn’t, we probably wouldn’t call it noise. However, don’t think that all aperiodic signals are random or unpredictable. For example, a monotonically increasing sequence \(x(t) = t\) is perfectly predictable, but it is also aperiodic because you never see the same value twice.

As a final example, a staircase signal, which is like the monotonically increasing sequence but is piece-wise constant, is also aperiodic even though you do see the same values multiple times.

There’s a real sense in which most signals are aperiodic. We typically use periodicity as a conceptual tool to understand idealized signals, but it’s also helpful to remember that any signal of finite duration can be made periodic by playing it on a loop. This idea will come back later when we get to the Fourier transform.

## 1.2.3. Fundamental frequency#

Periods are defined in units of time: how long must we wait before observing the signal repeating itself?
Often, it is more convenient to think in terms of *how many times* a signal repeats itself within a fixed duration of time, typically one second.

This idea provides a definition for **frequency**: how many cycles does a signal complete in one second?
Frequency is measured in units of Hertz (**Hz**), where 1 Hz denotes one full cycle in one second.

(Fundamental frequency)

If a signal \(x(t)\) has a fundamental period \(t_0\), then its **fundamental frequency** is defined as

Note that \(f_0\) need not be a whole number. In the pulse train example above, the period was \(1.5 = 3/2\) seconds, which is equivalent to \(f_0 = 2/3\) Hz: it completes two cycles in three seconds. (The third cycle starts at \(t=3\), as illustrated in the figure.)

If a signal is aperiodic, and has \(t_0 = \infty\), then its fundamental frequency is defined to be \(f_0 = 0\), meaning that it completes 0 cycles in 1 second, or any number of seconds. Note that this definition is primarily used for notational consistency — if a signal is aperiodic, it is also common to say that it has no fundamental frequency.

### 1.2.3.1. Why the \(t_0\) and \(f_0\) business?#

At this point, you might (rightly) be wondering why we have these 0 subscripts on “fundamental” values. There are two reasons for this:

To distinguish fundamental period (\(t_0\)) or frequency (\(f_0\)) from arbitrary times (\(t\)) or frequencies (\(f\)), and

To highlight the connection between the fundamental \(f_0\) and “harmonics” or “overtones”, which are often notated as \(f_1, f_2, \cdots\).

The first point is a somewhat arbitrary convention, but the second point is definitely not arbitrary. However, it’s important to keep in mind that any periodic signal will have a fundamental frequency, but we typically only discuss harmonics in the context of sinusoids.

This naturally leads us to the question: what is a wave?

## 1.2.4. Waves#

So far, we’ve discussed arbitrary signals in terms of their periodicity properties, and seen examples of periodic and aperiodic signals.
There are many kinds of periodic signals: pulse trains, square waves, triangle waves, sawtooth waves, just to name a few.
Among all periodic signals, **sinusoids** (e.g., sine and cosine waves like the figure below) are in many ways, the most important and mathematically well-behaved.
The term *wave* is therefore often used synonymously with the more specific *sinusoid*.

### 1.2.4.1. What makes sinusoids special?#

While many readers with some prior familiarity with audio have an intuitive grasp of sinusoids as “pure tones”, it’s often not clearly stated **why** sinusoids are so well-suited to signal processing and analysis, or where they come from in the first place.
The reasons underlying the use of sinusoidal waves go mathematically deep, but the basic principles can be understood directly in terms of geometry.

Keep in mind that our goal here is to understand periodicities in signals.
The physical universe readily supplies us with countless examples of periodic phenomena: think about the rotation of the Earth (day and night cycles), the orbit of the Moon around the Earth (full and new Moon cycles), the Earth orbiting the sun, and so on.
Each of these phenomena are *repetitive*.
Moreover, each of these phenomena are characterized by continuous **rotation**: once a full rotation has been completed, the cycle repeats itself.

Put succinctly, **rotation models repetition**.

### 1.2.4.2. Rotation and sinusoids#

Sinusoids are typically introduced in a high-school geometry class in the study of right triangles.
The sine of an angle \(\theta\) is defined as the ratio of the opposite side-length to the hypotenuse; the cosine being the ratio of the adjacent side-length to the hypotenuse, and so on.
While it is certainly useful in many contexts, this view can obscure the interpretation of sinusoids as *waves*.

Here, we’ll use a different, but equivalent definition of the sine (and cosine) of an angle. First, we will draw a circle with radius \(1\) centered at the origin (\(x=y=0\)). Next, we will draw a line through the origin and making an angle \(\theta\) with the horizontal axis. This line will intersect the circle at a point. Then:

\(\sin(\theta)\) is the height (distance up from the horizontal axis) of the point on the circle at angle \(\theta\);

\(\cos(\theta)\) is the width (distance right from the vertical axis) of the point on the circle at angle \(\theta\).

Note that when the circle has a radius of 1, the hypotenuse will always have length 1, so \(\sin\) and \(\cos\) measure the height and width of the triangle.

By convention, we take \(\theta=0\) to be the right-most point on the circle, the point \((x=1, y=0)\). Positive angles \(\theta > 0\) correspond to a **counter-clockwise** (upward) rotation from \((1, 0)\). Negative angles \(\theta < 0\) correspond to a **clockwise** (downward) rotation.

Remember: **sines and cosines turn angles into distances**.

**Why does this give us waves?**

The sine or cosine of a single angle just gives us a single number, between -1 and +1. A single number is not enough to get a wave—for that, we’ll need to change the angle over time. Imagine the angle varying continuously over time, like the seconds hand of a clock. We’ll denote this by \(\theta(t)\), and have \(\theta(0) = 0\) indicate that the starting position is at the right-most point on the circle \((1, 0)\).

Equivalently, we can think of this changing \(\theta(t)\) in terms of the \((x, y)\)-position of the point on the circle at the corresponding angle. To get a wave out of this continuous rotation, we can look at what happens to just one of the two coordinates this point, which will be given by

This process is illustrated by Fig. 1.8.