Overview

A course in 24 lessons. Start here.

The last time you heard someone's voice on the phone, you did not hear their voice.

What reached your ear was a reconstruction. Somewhere in the first millisecond, a codec decided your friend's vocal tract could be described by about ten numbers, refreshed fifty times a second, and threw away the waveform itself. It sent the description. The handset in your hand built a new voice from that description — never the original, close enough that you recognised who was speaking and heard that they were tired.

That is digital signal processing. Not a subject that gets applied to the world afterwards; the thing the world is already made of.

You already own the instrument

Put two fingers on the inside of your wrist and wait.

What you feel is a signal: a value that changes with time and carries information. It is nearly periodic, which means it has a frequency — call it 70 beats per minute, 1.2 Hz. But it is not exactly periodic, and the inexactness is the interesting part. The gaps between beats lengthen when you exhale and shorten when you inhale. Take those gaps, transform them, and a peak appears around 0.25 Hz that is your breathing, written into your pulse without your consent or knowledge. Below it, slower structure belonging to blood-pressure regulation. A clinician reads that plot and learns something about your nervous system that no single heartbeat contains.

You did not gain new data. The data was always there, in a form where it could not be seen. Changing the form is the whole trick, and it is available to you at the price of learning what a transform is.

That is the pattern this course teaches, over and over, in increasing depth: a signal arrives in the shape the physics happened to deliver it, and it is unreadable in that shape. You change the representation. The answer was already in the data.

Why bother mastering it

Plenty of skills are worth having. This one is worth more than it looks, for three reasons.

It transfers. The mathematics that separates a tone from noise in an audio recording separates a planet from starlight, a tumour boundary from tissue, a fatigue crack from machine hum, a heartbeat from a radar return through a wall. Not analogous mathematics — the same functions, called with different numbers. You learn convolution once. You spend the rest of your career recognising it in disguise.

It is falsifiable. A filter either removed the interference or it did not, and the residual says which. This is a field where you can be straightforwardly wrong and find out in a second, which is rarer and more valuable than it sounds. Most of what you will believe about a signal can be checked before you stake anything on it.

It is old enough to be deep and young enough to be unfinished. Fourier published in 1822. Cooley and Tukey made his transform fast in 1965 and changed what computers were for. Compressed sensing is younger than some people reading this, and it says something Nyquist would have called impossible. You are not arriving at a closed subject.

There is also the less defensible reason, which is the real one. Signals are beautiful. A spectrogram of a wren's song looks like handwriting. A filter's pole-zero plot tells you how it will sound before you hear it. At some point you stop seeing a plot of numbers and start seeing the mechanism that produced them, and that shift does not reverse.

What this course does differently

Most DSP teaching separates the idea from the machine. You derive something on a board on Tuesday, and on Thursday you fight an unrelated toolchain until it plots. The gap between those days is where people quit, and the quitting is usually blamed on the mathematics, which was not the problem.

Here the theory and the running code are the same paragraph.

Every lesson states an idea in plain terms, writes it as an equation you can read, and then does it — in Qu, in a few lines, with printed output you can reproduce on your own machine in under a second. No pseudo-code. No "an implementation is left to the reader." No screenshots of some other program. If a lesson claims a filter has 40 dB of stopband rejection, the lesson measures 40 dB in front of you, and if the number comes out at 38 the lesson says so and explains where the other 2 went.

You need no prior DSP. You need comfort with algebra, a rough memory of what sine and cosine do, and a working Qu install. Complex numbers get introduced properly when they are needed, in Lesson 4, as the tool they actually are rather than as a hazing ritual.

The map

Eight modules, twenty-four lessons. Read them in order the first time — each module is standing on the one before it.

Module 1 — Foundations. What a signal *is*, and the one property that makes a system tractable. *(1) Signal classification. (2) LTI systems and convolution. (3) Sampling and quantization.* By the end you will know why y[n] = Σ_k h[k] · x[n-k] describes almost every physical system you will meet, and why a wheel on film sometimes turns backwards.

Module 2 — Transforms and Spectra. The change of representation that makes the invisible visible. *(4) Fourier, DFT, FFT. (5) Laplace and Z-transforms. (6) Time-frequency and wavelets.* This is the module people remember. It contains the single most consequential algorithm in engineering and the reason your music streams at all.

Module 3 — Filters. Deciding what to keep. *(7) Analog and digital filters. (8) FIR and IIR design. (9) Adaptive filtering.* Ending with filters that tune themselves against an interference they were never told about — the mathematics inside every set of noise-cancelling headphones.

Module 4 — Random Signals. What to do when the signal will not repeat. *(10) Random processes and PSD. (11) Detection and estimation. (12) Wiener and Kalman filters.* Estimation under noise is where DSP stops being arithmetic and starts being inference. The Kalman filter flew to the Moon.

Module 5 — Communications. Getting a signal across a hostile gap. *(13) Modulation and demodulation. (14) Channels and equalization. (15) Coding and compression.* Including the result that says exactly how much information a noisy channel can carry, which is one of the few places engineering has a hard physical limit and knows its value.

Module 6 — Multidimensional DSP. Signals with more than one axis. *(16) Image and video processing. (17) Speech and audio. (18) Array and spatial processing.* A photograph is a signal in two dimensions, and sharpening it is a filter. Point enough microphones at a room and you can steer your hearing without moving anything.

Module 7 — Advanced Methods. *(19) Multirate DSP and filter banks. (20) Sparse and compressed sensing. (21) Machine learning for signals.* Lesson 20 reconstructs a signal from far fewer samples than the sampling theorem appears to permit. It is not a violation. It is a better question.

Module 8 — Implementation and Applications. Where the mathematics meets a finite machine. *(22) DSP hardware and fixed point. (23) Real-time embedded DSP. (24) Radar, biomedical, and control.* A filter that is correct in infinite precision and unstable in 16 bits is a bug with a specific name, and you will learn to see it coming.

That is the mountain. It is climbable in order, one lesson at a sitting.

A taste, before you go

You do not have to wait for Lesson 4 to see the central move of this entire subject. Three sounds played together — a note and two of its overtones, the recipe behind why a violin and a flute playing the same note are not the same sound:

fs = 8000                                  # 8 kHz sample rate

t  = (0 to 7999) / fs                      # one second of time

y  = sin(2*pi*440*t) + 0.5*sin(2*pi*880*t) + 0.25*sin(2*pi*1320*t)

As a list of 8000 numbers, y tells you nothing. It is a wobble. Now change the representation:

mag  = abs(rfft(y))                        # the spectrum

f    = (0 to len(mag) - 1) * fs / len(y)   # what each bin means, in Hz

peak = argmax(mag)
print("strongest component: {f[peak]:.1f} Hz")

strongest component: 440.0 Hz

strongest component: 440.0 Hz

Three ingredients went in. rfft found them, sorted by strength, without being told they were there, without being told anything about music. Plot mag against f and the three spikes stand alone at 440, 880 and 1320 Hz with nothing in between — the recipe recovered from the cake.

That is a one-second demonstration of the idea that took humanity until 1822 to state and until 1965 to make fast. Lesson 4 explains why it works. Lesson 1 starts three steps before it, with a question worth more than it first appears: what kind of signal is this, actually?

Next: Lesson 1 — Signal classification. Open a terminal first. This course is not a spectator sport.