Raymarching and Mandelbulbs

03 Feb 2025

Introduction

Fractals are some of the most mesmerizing visuals in computer graphics, but rendering them in 3D space requires special techniques beyond standard polygonal rendering. This article will take you into the world of ray marching, where we’ll use distance fields, lighting, and soft shadows to render a Mandelbulb fractal — one of the most famous 3D fractals.

By the end of this post, you’ll understand:

The basics of ray marching and signed distance functions (SDFs).
How to render 3D objects without polygons.
How to implement Phong shading for realistic lighting.
How to compute soft shadows for better depth.
How to animate a rotating Mandelbulb fractal.

This article will build on the knowledge that we established in the Basics of Shader Programming article that we put together earlier. If you haven’t read through that one, it’ll be worth taking a look at.

What is Ray Marching?

Ray marching is a distance-based rendering technique that is closely related to ray tracing. However, instead of tracing rays until they hit exact geometry (like in traditional ray tracing), ray marching steps along the ray incrementally using distance fields.

Each pixel on the screen sends out a ray into 3D space. We then march forward along the ray, using a signed distance function (SDF) to tell us how far we are from the nearest object. This lets us render smooth implicit surfaces like fractals and organic shapes.

Our first SDF

The simplest 3D object we can render using ray marching is a sphere. We define its shape using a signed distance function (SDF):

// Sphere Signed Distance Function (SDF)
float sdfSphere(vec3 p, float r) {
    return length(p) - r;
}

The sdfSphere() function returns the shortest distance from any point in space to the sphere’s surface.

We can now step along that ray until we reach the sphere. We do this by integrating our sfpSphere() function into our mainImage() function:

void mainImage(out vec4 fragColor, in vec2 fragCoord) {
    vec2 uv = (fragCoord - 0.5 * iResolution.xy) / iResolution.y;

    // Camera setup
    vec3 rayOrigin = vec3(0, 0, -3);  
    vec3 rayDir = normalize(vec3(uv, 1));

    float totalDistance = 0.0;
    const int maxSteps = 100;
    const float minDist = 0.001;
    const float maxDist = 20.0;
   
    for (int i = 0; i < maxSteps; i++) {
        vec3 pos = rayOrigin + rayDir * totalDistance;
        float dist = sdfSphere(pos, 1.0);

        if (dist < minDist) break;
        if (totalDistance > maxDist) break;

        totalDistance += dist;
    }

    vec3 col = (totalDistance < maxDist) ? vec3(1.0) : vec3(0.2, 0.3, 0.4);
    fragColor = vec4(col, 1.0);
}

First of all here, we convert the co-ordinate that we’re processing into screen co-ordinates:

vec2 uv = (fragCoord - 0.5 * iResolution.xy) / iResolution.y;

This uv value is now used to in the calculations to form our ray equation:

vec3 rayOrigin = vec3(0, 0, -3);  
vec3 rayDir = normalize(vec3(uv, 1));

We now iterate (march) down the ray to a maximum of maxSteps (currently set to 100) to determine if the ray intersects with our sphere (via sdfSphere).

Finally, we render the colour of our sphere if the distance is within tolerance; otherwise we consider this part of the background:

vec3 col = (totalDistance < maxDist) ? vec3(1.0) : vec3(0.2, 0.3, 0.4);
fragColor = vec4(col, 1.0);

You should see something similar to this:

Adding Lights

In order to make this sphere look a little more 3D, we can light it. In order to light any object, we need to be able to compute surface normals. We do that via a function like this:

vec3 getNormal(vec3 p) {
    vec2 e = vec2(0.001, 0.0);  // Small offset for numerical differentiation
    return normalize(vec3(
        sdfSphere(p + e.xyy, 1.0) - sdfSphere(p - e.xyy, 1.0),
        sdfSphere(p + e.yxy, 1.0) - sdfSphere(p - e.yxy, 1.0),
        sdfSphere(p + e.yyx, 1.0) - sdfSphere(p - e.yyx, 1.0)
    ));
}

We make decisions about the actual colour via a lighting function. This lighting function is informed by the surface normals that it computes:

vec3 lighting(vec3 p) {
    vec3 lightPos = vec3(2.0, 2.0, -2.0);  // Light source position
    vec3 normal = getNormal(p);  // Compute the normal at point 'p'
    vec3 lightDir = normalize(lightPos - p);  // Direction to light
    float diff = max(dot(normal, lightDir), 0.0);  // Diffuse lighting
    return vec3(diff);  // Return grayscale lighting effect
}

We can now integrate this back into our scene in the mainImage function. Rather than just making a static colour return when we establish a hit point, we start to execute the lighting function towards the end of the function:

void mainImage(out vec4 fragColor, in vec2 fragCoord) {
    vec2 uv = (fragCoord - 0.5 * iResolution.xy) / iResolution.y;

    // Camera setup
    vec3 rayOrigin = vec3(0, 0, -3);  // Camera positioned at (0,0,-3)
    vec3 rayDir = normalize(vec3(uv, 1));  // Forward-facing ray

    // Ray marching parameters
    float totalDistance = 0.0;
    const int maxSteps = 100;
    const float minDist = 0.001;
    const float maxDist = 20.0;
    vec3 hitPoint;
   
    // Ray marching loop
    for (int i = 0; i < maxSteps; i++) {
        hitPoint = rayOrigin + rayDir * totalDistance;
        float dist = sdfSphere(hitPoint, 1.0);  // Distance to the sphere

        if (dist < minDist) break;  // If we are close enough to the surface, stop
        if (totalDistance > maxDist) break;  // If we exceed max distance, stop

        totalDistance += dist;
    }

    // If we hit something, apply shading; otherwise, keep background color
    vec3 col = (totalDistance < maxDist) ? lighting(hitPoint) : vec3(0.2, 0.3, 0.4);
   
    fragColor = vec4(col, 1.0);
}

You should see something similar to this:

Mandelbulbs

We can now upgrade our rendering to use something a little more complex then our sphere.

SDF

The Mandelbulb is a 3D fractal inspired by the 2D Mandelbrot Set. Instead of working in 2D complex numbers, it uses spherical coordinates in 3D space.

The core formula: $z→zn+c$ is extended to 3D using spherical math.

Instead of a sphere SDF, we’ll use an iterative function to compute distances to the fractal surface.

float mandelbulbSDF(vec3 pos) {
    vec3 z = pos;
    float dr = 1.0;
    float r;
    const int iterations = 8;
    const float power = 8.0;

    for (int i = 0; i < iterations; i++) {
        r = length(z);
        if (r > 2.0) break;

        float theta = acos(z.z / r);
        float phi = atan(z.y, z.x);
        float zr = pow(r, power - 1.0);
        dr = zr * power * dr + 1.0;
        zr *= r;
        theta *= power;
        phi *= power;

        z = zr * vec3(sin(theta) * cos(phi), sin(theta) * sin(phi), cos(theta)) + pos;
    }

    return 0.5 * log(r) * r / dr;
}

This function iterates over complex numbers in 3D space to compute the Mandelbulb structure.

Raymarching the Mandelbulb

Now, we can take a look at what this produces. We use our newly created SDF to get our hit point. We’ll use this distance value as well to establish different colours.

void mainImage(out vec4 fragColor, in vec2 fragCoord) {
    vec2 uv = (fragCoord - 0.5 * iResolution.xy) / iResolution.y;

    // Camera setup
    vec3 rayOrigin = vec3(0, 0, -4);
    vec3 rayDir = normalize(vec3(uv, 1));

    // Ray marching parameters
    float totalDistance = 0.0;
    const int maxSteps = 100;
    const float minDist = 0.001;
    const float maxDist = 10.0;
    vec3 hitPoint;
   
    // Ray marching loop
    for (int i = 0; i < maxSteps; i++) {
        hitPoint = rayOrigin + rayDir * totalDistance;
        float dist = mandelbulbSDF(hitPoint);  // Fractal distance function

        if (dist < minDist) break;
        if (totalDistance > maxDist) break;

        totalDistance += dist;
    }

    // Color based on distance (simple shading)
    vec3 col = (totalDistance < maxDist) ? vec3(1.0 - totalDistance * 0.1) : vec3(0.1, 0.1, 0.2);

    fragColor = vec4(col, 1.0);
}

You should see something similar to this:

Rotation

We can’t see much with how this object is oriented. By adding some basic animation, we can start to look at the complexities of how this object is put together. We use the global iTime variable here to establish movement:

void mainImage(out vec4 fragColor, in vec2 fragCoord) {
    vec2 uv = (fragCoord - 0.5 * iResolution.xy) / iResolution.y;

    // Rotate camera around the fractal using iTime
    float angle = iTime * 0.5;  // Adjust speed of rotation
    vec3 rayOrigin = vec3(3.0 * cos(angle), 0.0, 3.0 * sin(angle)); // Circular path
    vec3 target = vec3(0.0, 0.0, 0.0); // Looking at the fractal
    vec3 forward = normalize(target - rayOrigin);
    vec3 right = normalize(cross(vec3(0, 1, 0), forward));
    vec3 up = cross(forward, right);
   
    vec3 rayDir = normalize(forward + uv.x * right + uv.y * up);

    // Ray marching parameters
    float totalDistance = 0.0;
    const int maxSteps = 100;
    const float minDist = 0.001;
    const float maxDist = 10.0;
    vec3 hitPoint;

    // Ray marching loop
    for (int i = 0; i < maxSteps; i++) {
        hitPoint = rayOrigin + rayDir * totalDistance;
        float dist = mandelbulbSDF(hitPoint);  // Fractal distance function

        if (dist < minDist) break;
        if (totalDistance > maxDist) break;

        totalDistance += dist;
    }

    // Color based on distance (simple shading)
    vec3 col = (totalDistance < maxDist) ? vec3(1.0 - totalDistance * 0.1) : vec3(0.1, 0.1, 0.2);

    fragColor = vec4(col, 1.0);
}

You should see something similar to this:

Lights

In order to make our fractal look 3D, we need to be able to compute our surface normals. We’ll be using the mandelbulbSDF function above to accomplish this:

vec3 getNormal(vec3 p) {
    vec2 e = vec2(0.001, 0.0);
    return normalize(vec3(
        mandelbulbSDF(p + e.xyy) - mandelbulbSDF(p - e.xyy),
        mandelbulbSDF(p + e.yxy) - mandelbulbSDF(p - e.yxy),
        mandelbulbSDF(p + e.yyx) - mandelbulbSDF(p - e.yyx)
    ));
}

We now use this function to light our object using phong shading:

// Basic Phong shading
vec3 phongLighting(vec3 p, vec3 viewDir) {
    vec3 normal = getNormal(p);

    // Light settings
    vec3 lightPos = vec3(2.0, 2.0, -2.0);
    vec3 lightDir = normalize(lightPos - p);
    vec3 ambient = vec3(0.1); // Ambient light

    // Diffuse lighting
    float diff = max(dot(normal, lightDir), 0.0);
   
    // Specular highlight
    vec3 reflectDir = reflect(-lightDir, normal);
    float spec = pow(max(dot(viewDir, reflectDir), 0.0), 16.0); // Shininess factor
   
    return ambient + diff * vec3(1.0, 0.8, 0.6) + spec * vec3(1.0); // Final color
}

Soft Shadows

To make the fractal look more realistic, we’ll implement soft shadows. This will really enhance how this object looks.

// Soft Shadows (traces a secondary ray to detect occlusion)
float softShadow(vec3 ro, vec3 rd) {
    float res = 1.0;
    float t = 0.02; // Small starting step
    for (int i = 0; i < 24; i++) {
        float d = mandelbulbSDF(ro + rd * t);
        if (d < 0.001) return 0.0; // Fully in shadow
        res = min(res, 10.0 * d / t); // Soft transition
        t += d;
    }
    return res;
}

Pulling it all together

We can now pull all of these enhancements together with our main image function:

void mainImage(out vec4 fragColor, in vec2 fragCoord) {
    vec2 uv = (fragCoord - 0.5 * iResolution.xy) / iResolution.y;

    // Rotating Camera
    float angle = iTime * 0.5;
    vec3 rayOrigin = vec3(3.0 * cos(angle), 0.0, 3.0 * sin(angle));
    vec3 target = vec3(0.0);
    vec3 forward = normalize(target - rayOrigin);
    vec3 right = normalize(cross(vec3(0, 1, 0), forward));
    vec3 up = cross(forward, right);
    vec3 rayDir = normalize(forward + uv.x * right + uv.y * up);

    // Ray marching
    float totalDistance = 0.0;
    const int maxSteps = 100;
    const float minDist = 0.001;
    const float maxDist = 10.0;
    vec3 hitPoint;

    for (int i = 0; i < maxSteps; i++) {
        hitPoint = rayOrigin + rayDir * totalDistance;
        float dist = mandelbulbSDF(hitPoint);

        if (dist < minDist) break;
        if (totalDistance > maxDist) break;

        totalDistance += dist;
    }

    // Compute lighting only if we hit the fractal
    vec3 color;
    if (totalDistance < maxDist) {
        vec3 viewDir = normalize(rayOrigin - hitPoint);
        vec3 baseLight = phongLighting(hitPoint, viewDir);
        float shadow = softShadow(hitPoint, normalize(vec3(2.0, 2.0, -2.0)));
        color = baseLight * shadow; // Apply shadows
    } else {
        color = vec3(0.1, 0.1, 0.2); // Background color
    }

    fragColor = vec4(color, 1.0);
}

Finally, you should see something similar to this:

Pretty cool!

The Basics of Shader Programming

02 Feb 2025

Introduction

Shaders are one of the most powerful tools in modern computer graphics, allowing real-time effects, lighting, and animation on the GPU (Graphics Processing Unit). They are used in games, simulations, and rendering engines to control how pixels and geometry appear on screen.

In this article, we’ll break down:

What shaders are and why they matter
How to write your first shader
Understanding screen coordinates
Animating a shader

This guide assumes zero prior knowledge of shaders and will explain each line of code step by step.

All of the code here can be run using Shadertoy. You won’t need to install any dependencies, but you will need a GPU-capable computer!

What is a Shader?

A shader is a small program that runs on the GPU. Unlike regular CPU code, shaders are executed in parallel for every pixel or vertex on the screen.

Types of Shaders

Vertex Shader – Moves and transforms individual points in 3D space.
Fragment Shader (Pixel Shader) – Determines the final color of each pixel.

For now, we’ll focus on fragment shaders since they control how things look.

Your First Shader

Let’s start with the simplest shader possible: a solid color fill.

Seeing Red!

void mainImage(out vec4 fragColor, in vec2 fragCoord) {
    fragColor = vec4(1.0, 0.0, 0.0, 1.0); // Solid red color
}

Breaking this code down:

void mainImage(...) → This function runs for every pixel on the screen.
fragColor → The output color of the pixel.
vec4(1.0, 0.0, 0.0, 1.0) → This defines an RGBA color:
- 1.0, 0.0, 0.0 → Red
- 1.0 → Fully opaque (no transparency)

Try This: Change the color values:

vec4(0.0, 1.0, 0.0, 1.0); → Green
vec4(0.0, 0.0, 1.0, 1.0); → Blue
vec4(1.0, 1.0, 0.0, 1.0); → Yellow

Mapping Colors to Screen Position

Instead of filling the screen with a single color, let’s map colors to pixel positions.

A Gradient Shader

void mainImage(out vec4 fragColor, in vec2 fragCoord) {
    vec2 uv = fragCoord / iResolution.xy; // Normalize coordinates (0 to 1)
    fragColor = vec4(uv.x, uv.y, 0.5, 1.0);
}

Breaking this code down:

fragCoord / iResolution.xy → Converts pixel coordinates into a 0 → 1 range.
uv.x → Controls red (left to right gradient).
uv.y → Controls green (bottom to top gradient).
0.5 → Keeps blue constant.

This creates a smooth gradient across the screen!

Try This: Swap uv.x and uv.y to see different patterns.

Animation

Shaders can react to time using iTime. This lets us create dynamic effects.

Moving Color Waves

void mainImage(out vec4 fragColor, in vec2 fragCoord) {
    vec2 uv = fragCoord / iResolution.xy;
    float wave = sin(uv.x * 10.0 + iTime);
    fragColor = vec4(wave, wave, wave, 1.0);
}

Breaking this code down:

sin(uv.x * 10.0 + iTime) → Creates waves that move over time.
The whole screen pulses from black to white dynamically.

Try This: Change 10.0 to 20.0 or 5.0 to make waves tighter or wider.

Wrapping up

Here have been some very simple shader programs to get started with. In future articles, we’ll build on this knowledge to build more exciting graphics applications.

A Primer on Radio Signals

30 Jan 2025

Introduction

Radio signals surround us every day—whether it’s FM radio in your car, WiFi on your laptop, or Bluetooth connecting your phone to wireless headphones. These signals are all based on the same fundamental principles: frequency, modulation, and bandwidth.

In today’s article, I want to go through some of the fundamentals:

How do radio signals travel through the air?
What is the difference between AM and FM?
How can multiple signals exist without interfering?
What is digital modulation (FSK & PSK), and why does it matter?
How can we capture and transmit signals using HackRF?

By the end of this article, we should be running some experiments with our own software defined radio devices.

Getting Setup

Before getting started in this next section, you’ll want to make sure that you have some specific software installed on your computer, as well as a software defined radio device.

I’m using a HackRF One as my device of choice as it integrates with all of the software that I use.

Make sure you have the following packages installed:

hackrf
gqrx
gnuradio

sudo pacman -S hackrf gqrx gnuradio

Create yourself a new python project and virtual environment, and install the libraries that will wrap some of these tools to give you an easy to use programming environment for your software defined radio.

# create your virtual environment
python -m venv venv

# activate
source venv/bin/activate

# install
pip install pyhackrf matplotlib

The output of some of these examples are graphs, so we use matplotlib to save them to look at later.

Be Responsible!

A note before we get going - you will be installing software that will allow you to transmit signals that could potentially be dangerous and against the law, so before transmitting:

Know the laws – Unlicensed transmission can interfere with emergency services.
Use ISM Bands – 433 MHz, 915 MHz, 2.4 GHz are allowed for low-power use.
Start in Receive Mode – Learning to capture first avoids accidental interference.

Basics of Radio Waves

What is a Radio Signal?

A radio signal is a type of electromagnetic wave that carries information through the air. These waves travel at the speed of light and can carry audio, video, or digital data.

Radio waves are defined by:

Frequency (Hz) – How fast the wave oscillates.
Wavelength (m) – The distance between peaks.
Amplitude (A) – The height of the wave (strength of the signal).

A high-frequency signal oscillates faster and has a shorter wavelength. A low-frequency signal oscillates slower and has a longer wavelength.

Since radio waves travel at the speed of light, their wavelength ($\lambda$) can be calculated using:

\[\lambda = \frac{c}{f}\]

Where:

$\lambda$ = Wavelength in meters
$c$ = Speed of light ~($3.0 \times 10^8$ m/s)
$f$ = Frequency in Hz

What is Frequency?

Frequency is measured in Hertz (Hz), meaning cycles per second. You may have heard of kilohertz, megahertz, and gigahertz. These are all common frequency units:

Unit	Hertz
1 kHz (kilohertz)	1,000 Hz
1 MHz (megahertz)	1,000,000 Hz
1 GHz (gigahertz)	1,000,000,000 Hz

Every device that uses radio has a specific frequency range. For example:

AM Radio: 530 kHz – 1.7 MHz
FM Radio: 88 MHz – 108 MHz
WiFi (2.4 GHz Band): 2.4 GHz – 2.5 GHz
Bluetooth: 2.4 GHz
GPS Satellites: 1.2 GHz – 1.6 GHz

Each of these frequencies belongs to the radio spectrum, which is carefully divided so that signals don’t interfere with each other.

What is Bandwidth?

Bandwidth is the amount of frequency space a signal occupies.

A narrowband signal (like AM radio) takes up less space. A wideband signal (like WiFi) takes up more space to carry more data.

Example:

AM Radio Bandwidth: ~10 kHz per station
FM Radio Bandwidth: ~200 kHz per station
WiFi Bandwidth: 20–80 MHz (much larger, more data)

The more bandwidth a signal has, the better the quality (or more data it can carry).

How Can Multiple Signals Exist Together?

One analogy you can use is imagining a highway—each lane is a different frequency. Cars (signals) stay in their lanes and don’t interfere unless they overlap (cause interference). This is why:

FM stations are spaced apart (88.1 MHz, 88.3 MHz, etc.).
WiFi has channels (1, 6, 11) to avoid congestion.
TV channels each have a dedicated frequency band.

This method of dividing the spectrum is called Frequency Division Multiplexing (FDM).

Using the following python code, we can visualise FDM in action by sweeping the FM spectrum:

# Basic FM Spectrum Capture

from hackrf import *
import matplotlib.pyplot as plt
import numpy as np
from scipy.signal import welch

with HackRF() as hrf:
    hrf.sample_rate = 20e6  # 20 MHz sample rate
    hrf.center_freq = 104.5e6  # FM Radio

    samples = hrf.read_samples(2e6)  # Capture 2 million samples

    # Compute PSD using Welch’s method (handling complex IQ data)
    freqs, psd_values = welch(samples, fs=hrf.sample_rate, nperseg=1024, return_onesided=False)

    # Convert frequency axis to MHz
    freqs_mhz = (freqs - (hrf.sample_rate / 2)) / 1e6 + (hrf.center_freq / 1e6)

    # Plot Power Spectral Density
    plt.figure(figsize=(10, 5))
    plt.plot(freqs_mhz, 10 * np.log10(psd_values))  # Convert power to dB
    plt.xlabel('Frequency (MHz)')
    plt.ylabel('Power Spectral Density (dB/Hz)')
    plt.title(f'FM Radio Spectrum at {hrf.center_freq/1e6} MHz')
    plt.grid()

    # Save and show
    plt.savefig("fm_spectrum.png")
    plt.show(block=True)

Running this code gave me the following resulting plot (it will be different for you depending on where you live!):

FM Spectrum

Each sharp peak that you see here represents an FM station at a unique frequency. These are the lanes.

Understanding Modulation

What is Modulation?

Radio signals don’t carry useful information by themselves. Instead, they use modulation to encode voice, music, or data.

There are two main types of modulation:

Analog Modulation – Used for traditional radio (AM/FM).
Digital Modulation – Used for WiFi, Bluetooth, GPS, and modern systems.

AM (Amplitude Modulation)

AM works by varying the height (amplitude) of the carrier wave to encode audio.

As an example, the carrier frequency stays the same (e.g., 900 kHz), but the amplitude changes based on the sound wave.

AM is prone to static noise (because any electrical interference changes amplitude).

You can capture a sample of AM signals using the hackrf_transfer utility that was installed on your system:

hackrf_transfer -r am_signal.iq -f 900000 -s 2000000

This will capture AM signals at 900 kHz into a file for later analysis.

We can write some python to capture an AM signal and plot the samples so we can visualise this information.

# AM Signal Demodulation

from hackrf import *
import matplotlib.pyplot as plt
import numpy as np

with HackRF() as hrf:
    hrf.sample_rate = 10e6  # 10 MHz sample rate
    hrf.center_freq = 693e6  # 693 MHz AM station

    samples = hrf.read_samples(1e6)  # Capture 1M samples

    # AM Demodulation - Extract Magnitude (Envelope Detection)
    demodulated = np.abs(samples)

    # Plot Demodulated Signal
    plt.figure(figsize=(10, 5))
    plt.plot(demodulated[:5000])  # Plot first 5000 samples
    plt.xlabel("Time")
    plt.ylabel("Amplitude")
    plt.title("AM Demodulated Signal")
    plt.grid()

    # Save and show
    plt.savefig("am_demodulated.png")
    plt.show(block=True)

Running this code should give you a plot of what’s happening at 693 MHz:

AM Demodulated

The plot above represents the amplitude envelope of a real AM radio transmission.

The X-axis represents time, while the Y-axis represents amplitude.
The variations in amplitude correspond to the audio signal encoded by the AM station.

FM (Frequency Modulation)

FM works by varying the frequency of the carrier wave to encode audio.

As an example, the amplitude stays constant, but the frequency changes based on the audio wave.

FM is clearer than AM because it ignores amplitude noise.

You can capture a sample of FM signals with the following:

hackrf_transfer -r fm_signal.iq -f 100300000 -s 2000000

This will capture an FM station at 100.3 MHz.

We can write some python code to capture and demodulate an FM signal as well:

from hackrf import *
import matplotlib.pyplot as plt
import numpy as np

with HackRF() as hrf:
    hrf.sample_rate = 2e6  # 2 MHz sample rate
    hrf.center_freq = 104.5e6  # Example FM station

    samples = hrf.read_samples(1e6)

    # FM Demodulation - Phase Differentiation
    phase = np.angle(samples)  # Extract phase
    fm_demodulated = np.diff(phase)  # Differentiate phase

    # Plot FM Demodulated Signal
    plt.figure(figsize=(10, 5))
    plt.plot(fm_demodulated[:5000])  # Plot first 5000 samples
    plt.xlabel("Time")
    plt.ylabel("Frequency Deviation")
    plt.title("FM Demodulated Signal")
    plt.grid()

    # Save and show
    plt.savefig("fm_demodulated.png")
    plt.show(block=True)

If you pick a frequency that has a local radio station, you should get a strong signal like this:

FM Demodulated

Unlike AM, where the signal’s amplitude changes, FM signals encode audio by varying the frequency of the carrier wave.

The graph above shows the frequency deviation over time:

The X-axis represents time, showing how the signal changes.
The Y-axis represents frequency deviation, showing how much the carrier frequency shifts.
The spikes and variations represent audio modulation, where frequency shifts encode sound.

If your FM demodulation appears too noisy:

Try tuning to a stronger station (e.g., 100.3 MHz).
Increase the sample rate for a clearer signal.
Apply a low-pass filter to reduce noise in post-processing.

Bandwidth of a Modulated Signal

Modulated signals require bandwidth ($B$), and the amount depends on the modulation type.

AM

The total bandwidth required for AM signals is:

\[B = 2f_m\]

Where:

$B$ = Bandwidth in Hz
$f_m$ = Maximum audio modulation frequency in Hz

If an AM station transmits audio up to 5 kHz, the bandwidth is:

\[B = 2 \times 5\text{kHz} = 10\text{kHz}\]

This explains why AM radio stations typically require ~10 kHz per station.

FM

The bandwidth required for an FM signal follows Carson’s Rule:

\[B = 2 (f_d + f_m)\]

Where:

$f_d$ = Peak frequency deviation (how much the frequency shifts)
$f_m$ = Maximum audio frequency in Hz

For an FM station with a deviation of 75 kHz and max audio frequency of 15 kHz, the total bandwidth is:

\[B = 2 (75 + 15) = 180 \text{kHz}\]

This explains why FM radio stations require much more bandwidth (~200 kHz per station).

Digital Modulation

For digital signals, it’s important to be able to transmit binary data (1’s and 0’s). These methods of modulation are focused on making this process much more optimal than what the analog counterparts could provide.

What is FSK (Frequency Shift Keying)?

FSK is digital FM—instead of smoothly varying frequency like FM radio, it switches between two frequencies for 0’s and 1’s. This method of modulation is used in technologies like Bluetooth, LoRa, and old-school modems.

Example:

A “0” might be transmitted as a lower frequency (e.g., 915 MHz).
A “1” might be transmitted as a higher frequency (e.g., 917 MHz).
The receiver detects these frequency changes and reconstructs the binary data.

What is PSK (Phase Shift Keying)?

PSK is digital AM—instead of changing amplitude, it shifts the phase of the wave. This method of modulation is used in technologies like WiFi, GPS, 4G LTE, Satellites.

Example:

0° phase shift = Binary 0
180° phase shift = Binary 1
More advanced PSK (like QPSK) uses four phase shifts (0°, 90°, 180°, 270°) to send two bits per symbol (faster data transmission).

Wrapping Up

In this post, we explored the fundamentals of radio signals—what they are, how they work, and how different modulation techniques like AM and FM allow signals to carry audio through the air.

This really is only the start of what you can get done with software defined radio. Here are some further resources to check out:

“The ARRL Handbook for Radio Communications” – A comprehensive guide to radio theory.
Great Scott Gadgets HackRF One – The official HackRF documentation.
RTL-SDR Blog – A great site for learning SDR with practical tutorials.
“SignalsEverywhere” – Video tutorials on SDR and radio hacking.

Getting Started with WebAssembly

29 Jan 2025

Introduction

Previously, we’ve explored WASM in rust as well as some more advanced concepts with Pixel Buffer Rendering again from Rust. In today’s article, we’ll go through WebAssembly from a more fundamental perspective.

WebAssembly (Wasm) is a powerful technology that enables high-performance execution in web browsers and beyond. If you’re just getting started, this guide will walk you through writing a simple WebAssembly program from scratch, running it in a browser using JavaScript.

What is WebAssembly?

WebAssembly is a low-level binary instruction format that runs at near-native speed. It provides a sandboxed execution environment, making it secure and highly portable. While it was initially designed for the web, Wasm is now expanding into cloud computing, serverless, and embedded systems.

Unlike JavaScript, Wasm allows near-native performance, making it ideal for gaming, video processing, and even AI in the browser.

First program

Before we start, we need to make sure all of the tools are available on your system. Make sure you have wabt installed on your system:

sudo pacman -S wabt

WAT

We’ll start by writing a WebAssembly module using the WebAssembly Text Format (WAT). Create a file called add.wat with the following code:

(module
  (func $add (param $a i32) (param $b i32) (result i32)
    local.get $a
    local.get $b
    i32.add
  )
  (export "add" (func $add))
)

This module defines a function $add that takes two 32-bit integers (i32) as parameters and returns their sum.

local.get retrieves the parameters.
i32.add performs the addition.
The function is exported as "add", making it accessible from JavaScript

wat2wasm

To convert our add.wat file into a .wasm binary, we’ll use a tool called wat2wasm from the WebAssembly Binary Toolkit (wabt) that we installed earlier:

wat2wasm add.wat -o add.wasm

This produces a binary add.wasm file, ready for execution.

Running WebAssembly from Javascript

Now, let’s create a JavaScript file (index.js) to load and execute our Wasm module:

async function runWasm() {
    // Fetch and compile the Wasm module
    const response = await fetch("add.wasm");
    const buffer = await response.arrayBuffer();
    const wasmModule = await WebAssembly.instantiate(buffer);

    // Get the exported add function
    const add = wasmModule.instance.exports.add;

    // Call the function
    console.log("5 + 7 = ", add(5, 7));
}

runWasm();

We can execute this javascript by referencing it from a html file, and running this in a browser.

Create index.html with the following:

<!DOCTYPE html>
<html lang="en">
<head>
    <meta charset="UTF-8">
    <meta name="viewport" content="width=device-width, initial-scale=1.0">
    <title>WebAssembly Example</title>
</head>
<body>
    <h1>WebAssembly Add Example</h1>
    <script src="index.js"></script>
</body>
</html>

Testing

Now we can serve our mini-website via python’s http server.

python3 -m http.server 8080

When you hit http://localhost:8080 in your browser, pop open your javascript console and you should see the following text:

5 + 7 = 12

Ok, it’s not the most exciting but it is the start of what you can achieve with these technologies.

Delaunay Triangulation

27 Jan 2025

Introduction

In the world of computational geometry, Delaunay triangulation stands out as one of the most versatile and powerful algorithms. Its ability to transform a scattered set of points into a structured mesh of triangles has applications ranging from terrain modeling to wireless network optimization.

This blog explores the concept of Delaunay triangulation, the algorithms that implement it, and its real-world applications.

What is Delaunay Triangulation?

Delaunay triangulation is a method for connecting a set of points in a plane (or higher dimensions) to form a network of triangles. The primary property of this triangulation is that no point lies inside the circumcircle of any triangle. This ensures the triangulation is “optimal” in the sense that it avoids skinny triangles and maximizes the smallest angles in the mesh.

A key relationship is its duality with the Voronoi diagram: Delaunay triangulation and Voronoi diagrams together provide complementary ways to describe the spatial relationships between points.

Why is Delaunay Triangulation Important?

The importance of Delaunay triangulation stems from its geometric and computational properties:

Optimal Mesh Quality: By avoiding narrow angles, it produces meshes suitable for simulations, interpolation, and rendering.
Simplicity and Efficiency: It reduces computational overhead by connecting points with minimal redundant edges.
Wide Applicability: From geographic information systems (GIS) to computer graphics and engineering, Delaunay triangulation plays a foundational role.

Real-World Applications

Geographic Information Systems (GIS)
- Terrain Modeling: Delaunay triangulation is used to create Triangulated Irregular Networks (TINs), which model landscapes by connecting elevation points into a mesh.
- Watershed Analysis: Helps analyze water flow and drainage patterns.
Computer Graphics
- Mesh Generation: Triangles are the fundamental building blocks for 3D modeling and rendering.
- Collision Detection: Used in simulations to detect interactions between objects.
Telecommunications
- Wireless Network Optimization: Helps optimize the placement of cell towers and the connections between them.
- Voronoi-based Coverage Analysis: Delaunay edges represent backhaul connections between towers.
Robotics and Pathfinding
- Motion Planning: Robots use triangulated graphs to navigate efficiently while avoiding obstacles.
- Terrain Navigation: Triangulation simplifies understanding of the environment for autonomous vehicles.
Engineering and Simulation
- Finite Element Analysis (FEA): Generates triangular meshes for simulating physical systems, such as stress distribution in materials.
- Fluid Dynamics: Simulates the flow of fluids over surfaces.
Environmental Science
- Flood Modeling: Simulates how water flows across landscapes.
- Resource Management: Models the distribution of natural resources like water or minerals.

A Practical Example

To illustrate the concept, let’s consider a set of points representing small towns scattered across a region. Using Delaunay triangulation:

The towns (points) are connected with lines (edges) to form a network.
These edges represent potential road connections, ensuring the shortest and most efficient routes between towns.
By avoiding sharp angles, this network is both practical and cost-effective for infrastructure planning.

Here’s a Python script that demonstrates this idea:

import numpy as np
import matplotlib.pyplot as plt
from scipy.spatial import Delaunay

# Generate random points in 2D space
np.random.seed(42)  # For reproducibility
points = np.random.rand(20, 2)  # 20 points in 2D

# Perform Delaunay triangulation
tri = Delaunay(points)

# Plot the points and the triangulation
plt.figure(figsize=(8, 6))
plt.triplot(points[:, 0], points[:, 1], tri.simplices, color='blue', linewidth=0.8)
plt.scatter(points[:, 0], points[:, 1], color='red', s=50, label='Points')
plt.title("Delaunay Triangulation Example")
plt.xlabel("X-axis")
plt.ylabel("Y-axis")
plt.legend()
plt.grid(True)
plt.show()

The following is the output from this program.

Example

Note how the paths between the points are the most optimal for connecting the towns efficiently. The triangulation avoids unnecessary overlaps or excessively sharp angles, ensuring practicality and simplicity in the network design.

Limitations and Challenges

While Delaunay triangulation is powerful, it has its challenges:

Degenerate Cases: Points that are collinear or on the same circle can cause issues.
Scalability: Large datasets may require optimized algorithms to compute triangulations efficiently.
Extensions to Higher Dimensions: In 3D or higher, the algorithm becomes more complex and computationally expensive.

Conclusion

Delaunay triangulation is a cornerstone of computational geometry, offering an elegant way to structure and connect scattered points. Its versatility makes it applicable across diverse domains, from GIS to robotics and environmental science. Whether you’re modeling terrains, optimizing networks, or simulating physical systems, Delaunay triangulation is an indispensable tool for solving real-world problems.

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Cogs and Levers A blog full of technical stuff

Raymarching and Mandelbulbs

Introduction

What is Ray Marching?

Our first SDF

Adding Lights

Mandelbulbs

SDF

Raymarching the Mandelbulb

Rotation

Lights

Soft Shadows

Pulling it all together

The Basics of Shader Programming

Introduction

What is a Shader?

Types of Shaders

Your First Shader

Seeing Red!

Mapping Colors to Screen Position

A Gradient Shader

Animation

Moving Color Waves

Wrapping up

A Primer on Radio Signals

Introduction

Getting Setup

Be Responsible!

Basics of Radio Waves

What is a Radio Signal?

What is Frequency?

What is Bandwidth?

How Can Multiple Signals Exist Together?

Understanding Modulation

What is Modulation?

AM (Amplitude Modulation)

FM (Frequency Modulation)

Bandwidth of a Modulated Signal

AM

FM

Digital Modulation

What is FSK (Frequency Shift Keying)?

What is PSK (Phase Shift Keying)?

Wrapping Up

Getting Started with WebAssembly

Introduction

What is WebAssembly?

First program

WAT

wat2wasm

Running WebAssembly from Javascript

Testing

Next

Delaunay Triangulation

Introduction

What is Delaunay Triangulation?

Why is Delaunay Triangulation Important?

Real-World Applications

A Practical Example

Limitations and Challenges

Conclusion