Constellation Diagrams – Part 1: The Complex Basics

A constellation diagram displays the symbol set for a given digital modulation scheme (BPSK, QPSK, 16-QAM, etc.). A symbol is a waveform—transmitted at some fixed interval—used to communicate one or more bits of data. The symbol set is the number of usable symbols possible within a specific modulation scheme. The size of the symbol set will determine the number of bits that each symbol can represent—the greater the number of available symbols, the greater the number of bits.

Each symbol is created by summing two sinusoid carriers—called the I and Q components—that are in quadrature, or 90° out of phase with each other. By convention, the I—or In-Phase component—will be a cosine wave, while the Q—or Quadrature component—will be a sine wave. The amplitude of each carrier can be one of several predetermined values, depending on the modulation scheme in use. Modifying the amplitude of either component will change the phase and amplitude of the resultant waveform, or symbol. The more potential amplitude levels a modulation scheme allows, the more unique symbols it can generate.

Constellation diagrams are a convenient way of quickly visualizing the phase and amplitude of each symbol, as well as the component waves from which they are constructed.

Basic Principles

A constellation diagram is formed by plotting these symbols as constellation points in the complex plane—using the amplitudes of the I and Q carriers as coordinates. The complex plane is similar to the familiar Cartesian xy‑plane with intersecting perpendicular axes; however, the horizontal axis represents the I component, and the vertical axis—the Q component. The amplitudes of each component carrier are depicted as either positive or negative numbers along the horizontal and vertical axes. Negative amplitude values indicate that the wave is inverted, or offset by 180°.

The angle, or phase, of each constellation point is measured counterclockwise from the horizontal axis. The amplitude of each symbol is measured as the distance from each constellation point to the origin of the chart where the two axes intersect.

In Figure 1 below, you’ll see a constellation diagram for 16-QAM. Notice that this diagram also includes the bit values that will be mapped to each symbol.

Figure 1 is an example of a square QAM constellation diagram. Square QAM constellations are typically used with symbol sets equal to an even power of two, such as 4-QAM (2²), 16-QAM (2⁴), 64-QAM (2⁶), etc.—and, as the name implies, consist of evenly spaced constellation points arranged in a square lattice. Although there are non-square QAM constellations—such as triangular QAM, cross QAM, hexagonal QAM, etc.—we will be focusing on the more common square QAM constellations, as they are the ones you will find in 802.11.

Constellation points are often represented as complex numbers. In the following sections, we’ll dive into what complex numbers are, how they relate to sinusoidal signals, and how to calculate the phase and amplitude of each modulated symbol.

Complex Numbers and Sinusoidal Signals

Mathematically, we can represent a sinusoidal signal as

$$ \large {\color{Green} A}\cos\left ( 2\pi f_{c}t+\theta \right ) $$

where $A$ is the amplitude, $2\pi f_{c}$ is the frequency of the carrier wave in Hertz (Hz), $t$ is time, and $θ$ is the phase. The carrier frequency will be dependent upon the communication channel in use, and the time period, or symbol length, is a fixed value determined by the modulation scheme, so we will consider them constants and focus on the amplitude and phase.

We know that the modulated symbol is a complex signal formed by summing the I (cosine) and Q (sine) components. So, it follows that we can represent the resultant waveform as:

$$ \large {\color{Green} A}\cos\left ( 2\pi f_{c}t+\theta \right )\:+\:{\color{Red} B}\cos\left ( 2\pi f_{c}t+\theta \right ) $$

Now, where do complex numbers fit into all of this?

Well, much like modulated symbols, complex numbers are also the sum of two components—a real part and an imaginary part. The “real” part is just a typical, everyday number, while the “imaginary” part will always be some multiple of the square root of ‑1, which we call the imaginary unit. This is denoted as the letter $i$ in mathematics, but in electrical engineering and its related fields, the letter $j$ is used instead, as the letter $i$ is already used to denote current.

$$ \large j=\sqrt{-1} $$

Complex numbers are often used in signal processing because they make certain mathematical calculations easier, or even possible, when working with sinusoids. They can tell us both the amplitude and phase of a given signal, making them well-suited for these tasks, but what makes them really compelling is a fundamental property of the imaginary unit itself:

Multiplying by the imaginary unit $j$ results in a rotation of 90°.*

*This rotational property of $j$ will be explained in more depth in a later section entitled “The Power of $j$.”

This key property of $j$ is what makes imaginary numbers—and therefore, complex numbers—so valuable in signal processing. It’s also what makes them ideal for representing constellation points, which are themselves the sum of two sinusoids, one in-phase and one offset by 90°—which brings us back to the formula:

$$ \large {\color{Green} A}\cos\left ( 2\pi f_{c}t+\theta \right )\:+\:{\color{Red} B}\cos\left ( 2\pi f_{c}t+\theta \right ) $$

We could represent this as the following complex number $z$—where the amplitude of the cosine component $\left ( A \right )$ becomes the real part $\left ( a \right )$, and the amplitude of the sine component $\left ( B \right )$ becomes the imaginary part $\left ( b \right )$.

$$ \large z={\color{Green} a}+j{\color{Red} b} $$

Any complex number can be expressed as an ordered pair in the form of $\left ( a, b \right )$, which we can use as coordinates in the complex plane—where the horizontal axis represents the real $\left ( Re \right )$ component, and the vertical axis represents the imaginary $\left ( Im \right )$ component. The resulting scatter plot is called an Argand diagram.

For example, let’s represent a waveform with the following complex number:

$$\large z={\color{Green} -3}+j{\color{Red} 3}$$

We can express this as the ordered pair (‑3, 3) and plot it as a point in the complex plane by first moving left along the $Re$ axis to ‑3, then up the $Im$ axis to +3.

Finding the Absolute Value

We will now determine the absolute value of this complex number, which we can define as the distance from the origin (0, 0) to the plotted point. When using complex numbers to represent waveforms, the absolute value will indicate the signal’s peak amplitude.

The absolute value of a number is the non-negative value of the number—ignoring its sign.

To better illustrate this, we will represent this complex number as a vector by drawing an arrow from the origin to the plotted point (‑3, 3). We will then draw a line from the plotted point back down to the $Re$ axis, forming a right triangle where the hypotenuse will be the vector’s magnitude. See Figure 3 below.

When dealing with vectors, the absolute value is referred to as the magnitude, but don’t be confused by the terminology—they both refer to the distance from the origin to the plotted point.

From here, we can simply calculate the absolute value using the Pythagorean Theorem. So, for the complex number $z$, the absolute value, denoted by the symbol $\left | {\color{Red} z} \right |$, will be:

$$ \large \left | {\color{Red} z} \right |= \sqrt{{\color{Green} a}^{2}\:+\:{\color{Red} b}^{2}} $$

This is illustrated in Figure 4 below.

Continuing with our example, let’s calculate the absolute value of $−3+j3$. Using the formula above, we get:

$$ \large \left | {\color{Red} z} \right | \:=\: \sqrt{\left ( {\color{Green} -3} \right )^{2} + {\color{Red} 3}^{2}} \:= \: \sqrt{9+9} \:= \: \sqrt{18} \:\approx\: {\color{Red} 4.243} $$

So, we can see that if we add a cosine wave and a sine wave with respective amplitudes of ‑3 and +3, the amplitude of the resultant wave will be 4.243 (after rounding to three decimal places).

Finding the Phase

Now that we’ve calculated the amplitude, let’s determine the signal’s phase. The phase is expressed as an angle and is measured counterclockwise from the $Re$ axis. See Figure 5 below, where the phase is depicted by the symbol ${\color{Purple} \theta }$.

To determine the angle of a complex number, we use the arctangent—or inverse tangent—function*; however, it gets a little tricky—as you’ll soon see.

*Don’t panic! You can find the inverse tangent function on almost any scientific calculator.

To find the phase of the complex number $a+jb$, we will use the following equation:

$$ \large {\color{Purple} \theta }= \tan^{-1}\left ( \frac{{\color{Red} b}}{{\color{Green} a}} \right ) $$

Accordingly, the phase of the signal represented by $−3+j3$, should be:

$$ \large {\color{Purple} \theta } \:=\: \tan^{-1}\left ( \frac{{\color{Red} 3}}{{\color{Green} -3}} \right ) \:=\: {\color{Purple} -45^{^{\circ}}} $$

That doesn’t look right, does it? To find the correct phase, we need to add 180° to our answer. This gives us 135°, which—if you refer back to Figure 5—seems more in line with what we would expect.

Now, this isn’t always the case. For example, let’s find the phase of the complex number $1+j1$. Using the equation above, we get the following:

$$ \large {\color{Purple} \theta } \:=\: \tan^{-1}\left ( \frac{{\color{Red} 1}}{{\color{Green} 1}} \right ) \:=\: {\color{Purple} 45^{^{\circ}}} $$

When we plot this number, you’ll see that this time, we do not have to add 180° to find the correct answer. See below.

So, what’s the trick?

The trick is knowing in which quadrant the number lies. If you look at Figure 7 below, you’ll notice that the intersecting axes of the Argand diagram forms four quadrants. They are numbered counterclockwise starting with Quadrant I in the upper right.

Here are the rules for determining the correct phase of a complex number:

• If the number is in Quadrant I, leave it alone.
• If the number is in either Quadrant II or Quadrant III, add 180°.
• If the number is in Quadrant IV, add 360°.

It’s easy to see in which quadrant a number resides just by plotting it, but what if you didn’t plot it? Is there a way to determine the quadrant just by looking at the complex number?

Yes.

For reference, let’s revisit the complex number $z$, where:

$$ \large z={\color{Green} a}+j{\color{Red} b} $$

We can determine the quadrant based on the following:

If ${\color{Green} a}$ is positive, then the number will be in either Quadrant I or Quadrant IV. To determine which one, we’ll need to look at ${\color{Red} b}$.

• If ${\color{Red} b}$ is positive, the number is in Quadrant I.
• If ${\color{Red} b}$ is negative, the number is in Quadrant IV.

If ${\color{Green} a}$ is negative, then the number will be in either Quadrant II or Quadrant III. To determine which one, we’ll need to look at ${\color{Red} b}$.

• If ${\color{Red} b}$ is positive, the number is in Quadrant II.
• If ${\color{Red} b}$ is negative, the number is in Quadrant III.

As an example, let’s determine the phase of the complex number:

$$ \large z={\color{Green} 3}-j{\color{Red} 3} $$

Following the guidelines above, we can see that since ${\color{Green} a}$ is positive, we know that this number will lie in either Quadrant I or Quadrant IV. To narrow it down further, we must look at the sign of ${\color{Red} b}$. Since ${\color{Red} b}$ is negative, we know that this number is in Quadrant IV.

So, plugging it into the phase equation stated earlier, we get:

$$ \large {\color{Purple} \theta } \:=\: \tan^{-1}\left ( \frac{{\color{Red} -3}}{{\color{Green} 3}} \right ) \:=\: {\color{Purple} -45^{^{\circ}}} $$

However, we’re not done. The rules above state that if the complex number resides in Quadrant IV, we need to add 360°.

$$ \large {\color{Purple} \theta } \:=\: -45^{^{\circ}} \:+\: 360^{^{\circ}} \:=\: {\color{Purple} 315^{^{\circ}}} $$

Let’s check our work by plotting $3−j3$.

Looking at Figure 8, it appears that we’ve successfully implemented the rules and found the correct phase.

The Power of j

Now that we’ve discussed complex numbers, plotted them, and determined their amplitude and phase, it’s time to revisit the imaginary unit $j$. As previously stated, multiplying by $j$ results in a rotation of 90°. This unique and important feature of the imaginary unit makes it a vital tool in the field of signal processing.

To illustrate this rotational property, let’s begin by calculating the powers of $j$—starting with $j^{0}$.

Considering any number to the power of zero is one, it follows that

$$ \large j^{0}=1 $$

Next, we’ll multiply $j^{0}$ by $j$. This results in $j^{1}$, which, of course, is just $j$.

$$ \large j^{1}=j $$

Multiplying $j^{1}$ by $j$ gives us $j^{2}$, and since $j$ is the square root of -1, it stands to reason that if we square $j$, we get ‑1.

$$ \large j^{2}=-1 $$

Then, we reach $j^{3}$. If we break this down into $j^{2} \times j^{1}$, we can clearly see that this becomes $-1 \times j$, or simply $−j$.

$$ \large j^{3}=-j $$

If we kept going, we would see that this pattern cycles over and over again, beginning with $j^{4}$ like so:

$$ \large j^{4}=1 $$ $$ \large j^{5}=j $$ $$ \large j^{6}=-1 $$ $$ \large j^{7}=-j $$

Now, let’s plot $j^{0}$ and $j^{1}$ as vectors in the complex plane. Since $j^{0}$ equals 1—which is a real number and has no imaginary component—we plot this directly on the $Re$ axis. Likewise, $j^{1}$ has no real component, so it is plotted directly on the $Im$ axis. See Figure 9 below.

We can clearly see that multiplying by $j$ rotates the vector 90°. Now, let’s plot $j^{2}$ and $j^{3}$ as well.

Figure 10 confirms that every multiplication of $j$ results in a rotation of 90°. This is what makes the imaginary unit so useful when representing and working with signals.

That’s it for now! In Part 2, we’ll discuss how to normalize constellations to the same average power using a scaling, or normalization factor.

If you want to learn more about the math behind complex numbers and signal processing, here are two excellent resources:

1. “The Scientist and Engineer’s Guide to Digital Signal Processing, copyright ©1997-1998 by Steven W. Smith. For more information visit the book’s website at: www.DSPguide.com“
2. https://wirelesspi.com/a-real-imaginative-guide-to-complex-numbers/