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

, 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:

1. Start with a real number, say

   1

   , on the positive Real Axis.

2. 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.

3. 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.

4. Multiply

   -1

   by j:

   -1 * j = -j

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

   -j

   on the negative Imaginary Axis.

5. 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:

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.

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!

Detection of tool wear during machining by designing a novel 12-way 2-shot learning model by applying L2-regularization and image augmentation – The International Journal of Advanced Manufacturing Technology

Tool wear monitoring is regarded as an incredibly important aspect of improving the surface integrity of machined components in the manufacturing sector. This research study performed operations using twelve different types of drilling and milling tools. The worn tools ranging from grade-1 to grade-5 were categorized based on tool wear severity by measuring the flank wear land width of each tool.

Photo by toolstop

When it comes to spectral analysis, I would also like to assert that Fourier analysis is going to be enough and sufficient in most of the cases. However, one may subscribe to other methods, such as wavelet analysis, as well. But what do all of these terms mean? Let’s not worry about them right now. We shall talk about simple stuff in the beginning.

In order to make life easier for my audience, I made a screencast of a famous song from Michael Jackson; Billie Jean. It is a well known pop song. And it has all sorts of drum beats, string twists of guitars, and other musical instruments. I am not really well versed with many western musical instruments. But I believe that this pop song is really going to serve my purpose. So, please click on the thumbnail below to open the song in a new tab of your browser window.

First of all I would like to draw your attention to the video. There is a spectral analyzer shown in the video. The analyzer has four boxes with graphs shown in them. As the video plays, you can notice the graphs beating in accordance with the videos. Let me explain these graphs a little bit first.

The graph in the upper left (UL) corner is known as a plot of the Fast Fourier Transform (FFT). Again don’t be confused with the fancy term. In really simple words it plots the different frequencies present in the audio (the song in our case) at any given point in time. It also gives us an idea as what the magnitudes of those frequencies is. What is noticeable about this is that there is white box in the top of the graph that has number in it, and it always keeps on changing its place. Moreover, please notice the way it changes its place. This box has the value of the frequency that has the highest value at any given point in time. As the frequency that has the highest magnitude can change from time to time, depending on what instrument or note is being played at any given point in time, this box keeps on shifting it position along the x-axis.

The way this box changes its position from time to time, remind me of the live dance of late Michael Jackson himself. It appears as if the box is dancing like Michael Jackson. Or at least, it definitely appears as if a dance is going on. Why is that? In my opinion the reason is that Michael wrote the song, composed its music, and choreographed his dance routines by putting all his heart and soul into it. So when he had to perform a live dance, he was moving his body and limbs just in perfect harmony with the changing notes. With every move of his body, he was capturing every change in the notes of the song.

But the white box is no dancer. It is an inanimate object. So why does it move in harmony with the beat of the song. Well, it does not. It is just an exhibition of the most prominent frequency in the song at any given point in time. And as that frequency changes, the box also changes its position. So it appears to be like a shadow of Michael.

What is the point of saying all of this. Let’s review rest of the boxes first. The box in the lower left (LL) corner is a similar plot but on an octave scale. Most music lovers really know what an octave scale is so I am not going to go in the details. To be brief, it is a logarithmic frequency scale. In essence it is also showing what the UL box is showing but on a different scale.

The box in the upper right (UR) corner is the spectrogram. Literally, this is also an exhibition of the spectral (frequcny) contents of the signal; the song in our case.

The last box is the one which is in lower right (LR) corner. This box has a single thread-like delineation in it. It fluctuates along with the music. This is what the music is doing as a function of time. You can think of it as the voltage fluctuation that happens as a function of the beat of the music. What is it anyways? The truth is that this box shows the overall loudness level of the song at any given moment in time. We shall come back to it later as it will help us in understanding various concepts in signal processing.

The last thing is the bar to the left of the analyzer. Now this bar literally shows ups and downs in perfect harmony with the ups and downs of the songs. This is also the loudness level of the song at any given moment in time.

The overall loudness, as the name suggests, is the sum total of the loudness of all the instruments being played in the music at any given point in time. There are different musical instruments involved in the creation of music. There is the sound of the drum. There is the audio of the guitar. And there is the voice of the vocalist. In all, there is a handful of instruments involved. Moreover, each one of them is possibly producing a sound of a different frequency from each other at every point in time. The sum total of the loudness of these sounds of different frequencies at any point in time is what we call as the overall loudness of the audio.Well, it need not be the sum total at all. It can be mean, or an aggregate of any kind. These aggregate loudness levels are shown in LR and the bar to the left of the analyzer.

So the overall loudness has all the information of the audio. We can see that its fluctuation is also in sync with the rhythm of the song. However, a huge problem with the overall loudness level is that it hides a lot of information about the audio. In case of music, we really do not know as to which notes are being played at any point in time. As a result, we remain in complete oblivion about the composition of the music.

Stereo systems actually use the overall loudness in the form of a time-varying voltages to render audible music through their speakers. Voltages are passed to the speakers. And they create audible sounds.

But what if we wanted to know the overall composition of the music? What instruments were played at a particular point in time, what were the notes, and how loud were they? Can we answer these questions? Fortunately the answer to these questions is yes. And we owe it to the genius of an eighteenth century French mathematician who went by the name of Jean-Baptiste Joseph Fourier who invented the famous Fourier transform. Fourier was a major in Napoleon’s army. However, his invention of this transform is bigger than all the military campaigns of the conqueror combined. Actually it would not be an exaggeration to say that Fourier transform has a greater impact than all the wars in history combined. We owe it to Fourier for most of the digital communication we observe around ourselves today.

According to Fourier, all the different frequencies that have been added together are separable. And Fourier transform is the tool to separate them. We see this exactly happening in LL, UL and UR boxes in the analyser. The Fourier transform takes as input what is shown in the LR bow. And creates spectra out of that; what is shown in LL, UL and UR. This is a remarkable contribution of Fourier analysis. As a result of applying it what we get is all the frequencies that make up the music and their respective magnitudes. And we get it for small segments of time. To this end, the Fourier analyzer takes a small segment of music at time (roughly around 25 milliseconds) and reveals the spectral information.

So what is the big deal of performing Fourier analysis. The big deal is that in applying the transform we went from knowing nothing about a seemingly interesting signal, to knowling everything about its spectral makeup. This is a tremendous achievement. Having this information in hand we can understand the mind of the musician as well as train a machine to understand that. Not having this information is tantamount to dealing with random electrical glitches. These glitches are shown in LR.

Photo by frans16611