Demystifying the “Imaginary”: How “j” and the Complex Plane Make DSP Click

black and white spiral staircase — Photo by Viktor Ruppert on Unsplash

If you’ve ever dipped your toes into Digital Signal Processing (DSP), you’ve probably encountered the mythical beast known as the “complex plane” and its equally enigmatic resident, the imaginary number $j$ (or $i$ in math circles). For many, these concepts are where DSP goes from “a bit tricky” to “utterly bewildering.”

But what if I told you that, for engineers, $j$ isn’t really “imaginary” at all? What if it’s actually one of the most practical, intuitive tools we have?

Richard Lyons, in his seminal “Understanding Digital Signal Processing,” has a gift for making these intimidating topics not just understandable, but obvious. Let’s unpack his wisdom and reveal the real-world power of $j$ and the complex plane.

Step 1: Beyond the Number Line – The Complex Plane

Forget the simple, one-dimensional number line we learned in grad school. That line is great for showing “how much,” but in DSP, we often need to know “how much” and “where it’s at” in a cycle. Think of a spinning wheel: you need to know its speed and its current position.

Enter the Complex Plane.

Instead of a line, imagine a two-dimensional grid:

The Horizontal Axis (Real Axis): This is your familiar number line. We use it to represent the “in-phase” part of a signal.
The Vertical Axis (Imaginary Axis): This is the new kid on the block. It represents the “quadrature” (or 90-degree shifted) part of a signal.

Now, instead of just a single number, any point on this plane is a complex number, made up of a real part and an imaginary part. We write it as

a + jb

, where

is the real part and

is the imaginary part.

Step 2: $j$ – Not Imaginary, But a Rotational Operator!

Here’s where Lyons flips the script. Stop thinking of $j$ as “the square root of -1” in a purely abstract sense. In DSP, $j$ has a concrete, visual job: it’s a 90-degree counter-clockwise rotation operator ¹.

Let’s see it in action:

Start with a real number, say

1

, on the positive Real Axis.
Multiply

1

by $j$ :

1 * j = j

. You’ve rotated

1

by 90 degrees counter-clockwise, and now you’re at

j

on the positive Imaginary Axis.
Multiply

j

by $j$ again:

j * j = j^2

. You’ve rotated another 90 degrees. You’re now at

-1

on the negative Real Axis.
Multiply

-1

by $j$ :

-1 * j = -j

. Another 90-degree rotation. You’re now at

-j

on the negative Imaginary Axis.
Multiply

-j

by $j$ :

-j * j = -j^2 = -(-1) = 1

. One more 90-degree rotation brings you back to

1

on the positive Real Axis.

See? $j$ isn’t making things disappear into some alternate reality. It’s simply turning things on a two-dimensional surface. This is the core insight that unlocks so much of DSP. This is the power of j. And j has brought us a long way.

Step 3: Spinning Signals – The Power of Phasors

Now, let’s bring it back to signals. Many real-world signals, like sound waves or radio waves, are sinusoidal—they oscillate up and down. Instead of describing them with messy sines and cosines, DSP engineers represent them as phasors: points that spin around the origin of the complex plane. Basically such waves do not oscillate up and down. They propagate from source to the sink in a helical manner.

The magic comes with Euler’s Equation:

e^{j\theta} = \cos(\theta) + j\sin(\theta)

This deceptively simple equation is the bedrock of modern DSP. It tells us that a complex exponential (the left side, which describes something spinning) is made up of a cosine wave (its projection onto the Real axis) and a sine wave (its projection onto the Imaginary axis).

Why is this a big deal?

Simplification: Multiplying complex exponentials (which involves just adding their angles) is far easier than wrestling with trigonometric identities. Want to shift a signal’s phase? Just add an angle to its complex representation.
Complete Picture: A single complex number can encapsulate both the amplitude (how big the signal is, represented by the length of the phasor) and the phase (where it is in its cycle, represented by its angle on the complex plane).

Step 4: The “Mystery” of Negative Frequency Solved

With the complex plane and spinning phasors in hand, “negative frequency” also becomes intuitive.

Imagine our phasor spinning.

Positive Frequency: The phasor spins counter-clockwise.
Negative Frequency: The phasor spins clockwise.

A real-world sine wave, the kind you measure with an oscilloscope, is actually the sum of two complex phasors: one spinning counter-clockwise (positive frequency) and one spinning clockwise (negative frequency). Their imaginary components cancel out, leaving only the real oscillation we observe.

\cos(\omega t) = \frac{e^{j\omega t} + e^{-j\omega t}}{2}

On a spectrum analyzer, when you see a spike at 100 Hz, you’ll also see a mirrored spike at -100 Hz for a real signal. They are two sides of the same coin, both necessary to describe that single real cosine wave.

The Practical Benefit: By converting real signals into purely complex (or “analytic”) signals through techniques like Hilbert transforms, we can eliminate one of these mirrored frequencies. This effectively doubles our usable bandwidth and simplifies many advanced DSP operations, especially in communications systems (like radio and cellular data).

The Takeaway

Richard Lyons strips away the fear surrounding “imaginary” numbers by showing their practical utility. The complex plane isn’t an abstract mathematical playground; it’s a two-dimensional whiteboard where we can draw and manipulate signals more effectively.

Understanding $j$ as a rotation, and grasping how phasors spin on this plane, unlocks the intuition behind powerful concepts like Euler’s Equation and negative frequency. These aren’t just theoretical constructs—they are the foundational tools that allow us to build everything from your smartphone to high-fidelity audio systems.

So, next time you see $j$ , don’t be intimidated. Just remember: it’s simply turning things around, making the complex world of signals a whole lot clearer!