The personal website of Scott W Harden
November 15th, 2022

Interfacing HX710 Differential ADC with Arduino

This page demonstrates how to read differential voltage from a HX710 ADC using Arduino. I recently obtained some pressure sensor boards from Amazon for less than $3 each under names like 6pcs 3.3-5V Digital Barometric Air Pressure Sensor Module Liquid Water Level Controller Board 0-40KPa that use this ADC. Several years ago I worked on a precision pressure meter project based on an I2C temperature and pressure sensor (MS5611), and now that I see new inexpensive SPI pressure sensor modules on the consumer market I'm interested to learn more about their capabilities.

Analog-to-Digital Converter IC

The ADC chip is easily identified as a HX710B 24-Bit Analog-to-Digital Converter (ADC) with Built-in Temperature Sensor. According to the datasheet it can be powered by a 3.3V or 5V supply, and the value it reports is the differential voltage between two input pins.

The datasheet indicates this device can be run from a 3.3V or 5V supply, it uses a built-in fixed-gain (128x) differential amplifier, and it can read up to 40 samples per second. The datasheet provides an example circuit demonstrating how this ADC can be used to measure weight from a scale sensor:

Pressure Sensor

To get a better idea of how this sensor works it would be helpful to locate its product number. I had a hunch it was beneath the part so I desoldered it, and indeed I found part identification information.

The pressure sensor is labeled as a PSG010S but unfortunately I struggled to find a quality datasheet for it. I did find some now-deleted images from an AliExpress listing showing the differences between the base model and the R and S variants. I found this PSG010R datasheet (curiously written in Comic Sans) indicating that maximum voltage is 5V and that the gauge pressure is 0 - 40KPa (0 - 5.8 PSI). This seems to be a fairly standard differential pressure sensor design using a pair of voltage dividers where the pressure is a function of the difference in voltage at the two mid-points (a Wheatstone bridge).

Read HX710B with Arduino

This code demonstrates how to measure HX710B values using Arduino and display the readings in the serial terminal sufficient to graph in real time using the serial plotter. The animated plot is what it looks like when I blow puffs of air on the sensor.

const int HX_OUT_PIN = 2;
const int HX_SCK_PIN = 3;

enum HX_MODE { NONE, DIFF_10Hz, TEMP_40Hz, DIFF_40Hz};
const byte HX_MODE = DIFF_40Hz;

void setup() {
  pinMode(HX_SCK_PIN, OUTPUT);
  pinMode(HX_OUT_PIN, INPUT);
  Serial.begin(9600);
}

void loop() {
  Serial.println(readHX());
}

unsigned long readHX() {

  // pulse clock line to start a reading
  for (char i = 0; i < HX_MODE; i++) {
    digitalWrite(HX_SCK_PIN, HIGH);
    digitalWrite(HX_SCK_PIN, LOW);
  }

  // wait for the reading to finish
  while (digitalRead(HX_OUT_PIN)) {}

  // read the 24-bit pressure as 3 bytes using SPI
  byte data[3];
  for (byte j = 3; j--;) {
    data[j] = shiftIn(HX_OUT_PIN, HX_SCK_PIN, MSBFIRST);
  }

  data[2] ^= 0x80;  // flip the 'out of range' error bit

  // shift the 3 bytes into a large integer
  long result;
  result += (long)data[2] << 16;
  result += (long)data[1] << 8;
  result += (long)data[0];

  return result;
}

Open-Source HX710B Libraries

Although some libraries are available which facilitate interacting with the HX710, here I engage with it discretely to convey each step of the conversion and measurement process. I found that many libraries use the 10 Hz mode by default, whereas I certainly prefer the 40 Hz mode. More frustratingly, code in many libraries refer to this as gain, which is incorrect. The datasheet indicates gain is fixed at 128 and cannot be changed in software.

Resources

Markdown source code last modified on November 17th, 2022
---
Title: Interfacing HX710 Differential ADC with Arduino
Description: How to read differential voltage from a HX710 ADC using Arduino
Date: 2022-11-15 12:30AM EST
Tags: circuit, microcontroller
---

# Interfacing HX710 Differential ADC with Arduino

**This page demonstrates how to read differential voltage from a HX710 ADC using Arduino.** I recently obtained some pressure sensor boards from Amazon for less than $3 each under names like _6pcs 3.3-5V Digital Barometric Air Pressure Sensor Module Liquid Water Level Controller Board 0-40KPa_ that use this ADC. Several years ago I worked on a [precision pressure meter project](https://swharden.com/blog/2017-04-29-precision-pressure-meter-project/) based on an [I2C](https://en.wikipedia.org/wiki/I%C2%B2C)  temperature and pressure sensor ([MS5611](https://www.te.com/commerce/DocumentDelivery/DDEController?Action=showdoc&DocId=Data+Sheet%7FMS5611-01BA03%7FB3%7Fpdf%7FEnglish%7FENG_DS_MS5611-01BA03_B3.pdf%7FCAT-BLPS0036)), and now that I see new inexpensive [SPI](https://en.wikipedia.org/wiki/Serial_Peripheral_Interface) pressure sensor modules on the consumer market I'm interested to learn more about their capabilities.

<img src="hx710b-pressure-board.jpg" class="my-5 border border-dark shadow img-fluid w-75 mx-auto d-block">

## Analog-to-Digital Converter IC

**The ADC chip is easily identified as a [HX710B](https://www.electronicscomp.com/datasheet/hx710b-ic-datasheet.pdf) _24-Bit Analog-to-Digital Converter (ADC) with Built-in Temperature Sensor_.** According to the datasheet it can be powered by a 3.3V or 5V supply, and the value it reports is the differential voltage between two input pins. 

<img src="hx710b-pinout.jpg" class="my-5 img-fluid w-75 mx-auto d-block">

The datasheet indicates this device can be run from a 3.3V or 5V supply, it uses a built-in fixed-gain (128x) differential amplifier, and it can read up to 40 samples per second. The datasheet provides an example circuit demonstrating how this ADC can be used to measure weight from a scale sensor:

<img src="hx710-datasheet.jpg" class="my-5 img-fluid w-75 mx-auto d-block">

## Pressure Sensor

To get a better idea of how this sensor works it would be helpful to locate its product number. I had a hunch it was beneath the part so I desoldered it, and indeed I found part identification information.

<img src="hx710b-pressure-psg010s.jpg" class="my-5 border border-dark shadow img-fluid w-75 mx-auto d-block">

**The pressure sensor is labeled as a PSG010S** but unfortunately I struggled to find a quality datasheet for it. I did find some now-deleted images from an AliExpress listing showing the differences between the base model and the R and S variants. 
I found [this PSG010R datasheet](https://www.katranji.com/tocimages/files/536845-544144.pdf) (curiously written in Comic Sans) indicating that maximum voltage is 5V and that the gauge pressure is 0 - 40KPa (0 - 5.8 PSI). This seems to be a fairly standard [differential pressure sensor](https://www.avnet.com/wps/portal/abacus/solutions/technologies/sensors/pressure-sensors/measurement-types/differential/) design using a pair of voltage dividers where the pressure is a function of the difference in voltage at the two mid-points (a [Wheatstone bridge](https://en.wikipedia.org/wiki/Wheatstone_bridge)).


<img src="psg-pressure-sensor.jpg" class="my-5 border border-dark shadow img-fluid w-75 mx-auto d-block">

## Read HX710B with Arduino

**This code demonstrates how to measure HX710B values using Arduino** and display the readings in the serial terminal sufficient to graph in real time using the serial plotter. The animated plot is what it looks like when I blow puffs of air on the sensor.

<img src="hx710-arduino-plot.gif" class="my-5 img-fluid mx-auto d-block">

```c
const int HX_OUT_PIN = 2;
const int HX_SCK_PIN = 3;

enum HX_MODE { NONE, DIFF_10Hz, TEMP_40Hz, DIFF_40Hz};
const byte HX_MODE = DIFF_40Hz;

void setup() {
  pinMode(HX_SCK_PIN, OUTPUT);
  pinMode(HX_OUT_PIN, INPUT);
  Serial.begin(9600);
}

void loop() {
  Serial.println(readHX());
}

unsigned long readHX() {

  // pulse clock line to start a reading
  for (char i = 0; i < HX_MODE; i++) {
    digitalWrite(HX_SCK_PIN, HIGH);
    digitalWrite(HX_SCK_PIN, LOW);
  }

  // wait for the reading to finish
  while (digitalRead(HX_OUT_PIN)) {}

  // read the 24-bit pressure as 3 bytes using SPI
  byte data[3];
  for (byte j = 3; j--;) {
    data[j] = shiftIn(HX_OUT_PIN, HX_SCK_PIN, MSBFIRST);
  }
  
  data[2] ^= 0x80;  // flip the 'out of range' error bit

  // shift the 3 bytes into a large integer
  long result;
  result += (long)data[2] << 16;
  result += (long)data[1] << 8;
  result += (long)data[0];

  return result;
}
```

## Open-Source HX710B Libraries

Although some libraries are available which facilitate interacting with the HX710, here I engage with it discretely to convey each step of the conversion and measurement process. I found that many libraries use the 10 Hz mode by default, whereas I certainly prefer the 40 Hz mode. More frustratingly, code in many libraries refer to this as _gain_, which is incorrect. The datasheet indicates gain is fixed at 128 and cannot be changed in software.

## Resources

* [Pressure Sensor Guide](https://www.electroschematics.com/pressure-sensor-guide/) by T.K. HAREENDRAN - A similar write-up that goes into additional detail. They didn't de-solder the pressure sensor to identify the component name, but there's lots of good information on this page.

* [bogde/HX711 on GitHub](https://github.com/bogde/HX711) - An Arduino library to interface the Avia Semiconductor HX711 24-Bit Analog-to-Digital Converter (ADC) for reading load cells / weight scales. Code on this page does not use this library, but others may find it helpful.

* [HX710 datasheet (English)](https://www.electronicscomp.com/datasheet/hx710b-ic-datasheet.pdf)

* [Differential pressure sensors](https://www.avnet.com/wps/portal/abacus/solutions/technologies/sensors/pressure-sensors/measurement-types/differential/) - Article about the topic which includes a good example of an instrumentation amplifier.

* [Design tips for a resistive-bridge pressure sensor in industrial process-control systems](https://www.ti.com/lit/an/slyt640/slyt640.pdf) - Texas Instruments application note

* [The Wheatstone Bridge](https://meritsensor.com/the-wheatstone-bridge/) by Michael Daily
November 7th, 2022

Creating Bitmaps from Scratch in C#

This project how to represent bitmap data in a plain old C object (POCO) to create images from scratch using C# and no dependencies. Common graphics libraries like SkiaSharp, ImageSharp, System.Drawing, and Maui.Graphics can read and write bitmaps in memory, so a POCO that stores image data and converts it to a bitmap byte allows creation of platform-agnostic APIs that can be interfaced from any graphics library.

This page demonstrates how to use C# (.NET 6.0) to create bitmap images from scratch. Bitmap images can then be saved to disk and viewed with any image editing program, or they can consumed as a byte array in memory by a graphics library. There are various bitmap image formats (grayscale, indexed colors, 16-bit, 32-bit, transparent, etc.) but code here demonstrates the simplest common case (8-bit RGB color).

Representing Color

The following struct represents RGB color as 3 byte values and has helper methods for creating new colors.

public struct RawColor
{
    public readonly byte R, G, B;

    public RawColor(byte r, byte g, byte b)
    {
        (R, G, B) = (r, g, b);
    }

    public static RawColor Random(Random rand)
    {
        byte r = (byte)rand.Next(256);
        byte g = (byte)rand.Next(256);
        byte b = (byte)rand.Next(256);
        return new RawColor(r, g, b);
    }

    public static RawColor Gray(byte value)
    {
        return new RawColor(value, value, value);
    }
}

A color class like this could be extended to support additional niceties. Refer to SkiaSharp's SKColor.cs, System.Drawing's Color.cs, and Maui.Graphics' Color.cs for examples and implementation details. I commonly find the following features useful include when writing a color class:

  • A static class with named colors e.g., RawColors.Blue
  • Conversion to/from ARGB e.g., RawColor.FromAGRB(123456)
  • Conversion to/from HTML e.g., RawColor.FromHtml(#003366)
  • Conversion between RGB and HSL/HSV
  • Helper functions to Lighten() and Darken()
  • Helper functions to ShiftHue()
  • Extension methods to convert to common other formats like SKColor

Representing the Bitmap Image

This is the entire image class and it serves a few specific roles:

  • Store image data in a byte array arranged identically to how it will be exported in the bitmap
  • Provide helper methods to get/set pixel color
  • Provide a method to return the image as a bitmap by adding a minimal header
public class RawBitmap
{
    public readonly int Width;
    public readonly int Height;
    private readonly byte[] ImageBytes;

    public RawBitmap(int width, int height)
    {
        Width = width;
        Height = height;
        ImageBytes = new byte[width * height * 4];
    }

    public void SetPixel(int x, int y, RawColor color)
    {
        int offset = ((Height - y - 1) * Width + x) * 4;
        ImageBytes[offset + 0] = color.B;
        ImageBytes[offset + 1] = color.G;
        ImageBytes[offset + 2] = color.R;
    }

    public byte[] GetBitmapBytes()
    {
        const int imageHeaderSize = 54;
        byte[] bmpBytes = new byte[ImageBytes.Length + imageHeaderSize];
        bmpBytes[0] = (byte)'B';
        bmpBytes[1] = (byte)'M';
        bmpBytes[14] = 40;
        Array.Copy(BitConverter.GetBytes(bmpBytes.Length), 0, bmpBytes, 2, 4);
        Array.Copy(BitConverter.GetBytes(imageHeaderSize), 0, bmpBytes, 10, 4);
        Array.Copy(BitConverter.GetBytes(Width), 0, bmpBytes, 18, 4);
        Array.Copy(BitConverter.GetBytes(Height), 0, bmpBytes, 22, 4);
        Array.Copy(BitConverter.GetBytes(32), 0, bmpBytes, 28, 2);
        Array.Copy(BitConverter.GetBytes(ImageBytes.Length), 0, bmpBytes, 34, 4);
        Array.Copy(ImageBytes, 0, bmpBytes, imageHeaderSize, ImageBytes.Length);
        return bmpBytes;
    }

    public void Save(string filename)
    {
        byte[] bytes = GetBitmapBytes();
        File.WriteAllBytes(filename, bytes);
    }
}

Generate Images from Scratch

The following code uses the bitmap class and color struct above to create test images

Random Colors

RawBitmap bmp = new(400, 300);
Random rand = new();
for (int x = 0; x < bmp.Width; x++)
    for (int y = 0; y < bmp.Height; y++)
        bmp.SetPixel(x, y, RawColor.Random(rand));
bmp.Save("random-rgb.bmp");

Rainbow

RawBitmap bmp = new(400, 300);
Random rand = new();
for (int x = 0; x < bmp.Width; x++)
{
    for (int y = 0; y < bmp.Height; y++)
    {
        byte r = (byte)(255.0 * x / bmp.Width);
        byte g = (byte)(255.0 * y / bmp.Height);
        byte b = (byte)(255 - 255.0 * x / bmp.Width);
        RawColor color = new(r, g, b);
        bmp.SetPixel(x, y, color);
    }
}
bmp.Save("rainbow.bmp");

Rectangles

RawBitmap bmp = new(400, 300);
Random rand = new();
for (int i = 0; i < 1000; i++)
{
    int rectX = rand.Next(bmp.Width);
    int rectY = rand.Next(bmp.Height);
    int rectWidth = rand.Next(50);
    int rectHeight = rand.Next(50);
    RawColor color = RawColor.Random(rand);

    for (int x = rectX; x < rectX + rectWidth; x++)
    {
        for (int y = rectY; y < rectY + rectHeight; y++)
        {
            if (x < 0 || x >= bmp.Width) continue;
            if (y < 0 || y >= bmp.Height) continue;
            bmp.SetPixel(x, y, color);
        }
    }
}
bmp.Save("rectangles.bmp");

Interfacing Graphics Libraries

The following code demonstrates how to load the bitmap byte arrays generated above into common graphics libraries and save the result as a JPEG file. Although the bitmap byte array can be written directly to disk as a .bmp file, these third-party libraries are required to encode images in additional formats like JPEG.

System.Drawing

using System.Drawing;

static void SaveBitmap(byte[] bytes, string filename = "demo.jpg")
{
    using MemoryStream ms = new(bytes);
    using Image img = Bitmap.FromStream(ms);
    img.Save(filename);
}

SkiaSharp

using SkiaSharp;

static void SaveBitmap(byte[] bytes, string filename = "demo.jpg")
{
    using SKBitmap bmp = SKBitmap.Decode(bytes);
    using SKFileWStream fs = new(filename);
    bmp.Encode(fs, SKEncodedImageFormat.Jpeg, quality: 95);
}

ImageSharp

using SixLabors.ImageSharp;
using SixLabors.ImageSharp.Formats.Jpeg;

static void SaveBitmap(byte[] bytes, string filename = "demo.jpg")
{
    using Image img = Image.Load(bytes);
    JpegEncoder encoder = new() { Quality = 95 };
    img.Save(filename, encoder);
}

Resources

Markdown source code last modified on November 8th, 2022
---
Title: Creating Bitmaps from Scratch in C#
Description: How to create a bitmap and set pixel colors in memory and save the result to disk or convert it to a traditional image format
Date: 2022-11-07 18:45PM EST
Tags: csharp, graphics
---

# Creating Bitmaps from Scratch in C# 

**This project how to represent bitmap data in a plain old C object (POCO) to create images from scratch using C# and no dependencies.** 
Common graphics libraries like [SkiaSharp](https://swharden.com/csdv/skiasharp/), [ImageSharp](https://swharden.com/csdv/platforms/imagesharp/), [System.Drawing](https://swharden.com/csdv/system.drawing/), and [Maui.Graphics](https://swharden.com/csdv/maui.graphics/) can read and write bitmaps in memory, so a [POCO](https://en.wikipedia.org/wiki/POCO) that stores image data and converts it to a bitmap byte allows creation of platform-agnostic APIs that can be interfaced from any graphics library.

**This page demonstrates how to use C# (.NET 6.0) to create bitmap images from scratch.** Bitmap images can then be saved to disk and viewed with any image editing program, or they can consumed as a byte array in memory by a graphics library. There are various [bitmap image formats](https://learn.microsoft.com/en-us/dotnet/desktop/winforms/advanced/types-of-bitmaps?view=netframeworkdesktop-4.8) (grayscale, indexed colors, 16-bit, 32-bit, transparent, etc.) but code here demonstrates the simplest common case (8-bit RGB color).

## Representing Color

The following `struct` represents RGB color as 3 `byte` values and has helper methods for creating new colors. 

```cs
public struct RawColor
{
    public readonly byte R, G, B;

    public RawColor(byte r, byte g, byte b)
    {
        (R, G, B) = (r, g, b);
    }

    public static RawColor Random(Random rand)
    {
        byte r = (byte)rand.Next(256);
        byte g = (byte)rand.Next(256);
        byte b = (byte)rand.Next(256);
        return new RawColor(r, g, b);
    }

    public static RawColor Gray(byte value)
    {
        return new RawColor(value, value, value);
    }
}
```

A color class like this could be extended to support additional niceties. Refer to [SkiaSharp's `SKColor.cs`](https://github.com/mono/SkiaSharp/blob/main/binding/Binding/SKColor.cs), [System.Drawing's `Color.cs`](https://github.com/dotnet/runtime/blob/main/src/libraries/System.Drawing.Primitives/src/System/Drawing/Color.cs), and [Maui.Graphics' `Color.cs`](https://github.com/dotnet/maui/blob/main/src/Graphics/src/Graphics/Color.cs) for examples and implementation details. I commonly find the following features useful include when writing a color class:

* A static class with named colors e.g., `RawColors.Blue`
* Conversion to/from ARGB e.g., `RawColor.FromAGRB(123456)`
* Conversion to/from HTML e.g., `RawColor.FromHtml(#003366)`
* Conversion between RGB and [HSL/HSV](https://en.wikipedia.org/wiki/HSL_and_HSV)
* Helper functions to `Lighten()` and `Darken()`
* Helper functions to `ShiftHue()`
* Extension methods to convert to common other formats like `SKColor`

## Representing the Bitmap Image

This is the entire image class and it serves a few specific roles:

* Store image data in a byte array arranged identically to how it will be exported in the bitmap
* Provide helper methods to get/set pixel color
* Provide a method to return the image as a bitmap by adding a minimal header

```cs
public class RawBitmap
{
    public readonly int Width;
    public readonly int Height;
    private readonly byte[] ImageBytes;

    public RawBitmap(int width, int height)
    {
        Width = width;
        Height = height;
        ImageBytes = new byte[width * height * 4];
    }

    public void SetPixel(int x, int y, RawColor color)
    {
        int offset = ((Height - y - 1) * Width + x) * 4;
        ImageBytes[offset + 0] = color.B;
        ImageBytes[offset + 1] = color.G;
        ImageBytes[offset + 2] = color.R;
    }

    public byte[] GetBitmapBytes()
    {
        const int imageHeaderSize = 54;
        byte[] bmpBytes = new byte[ImageBytes.Length + imageHeaderSize];
        bmpBytes[0] = (byte)'B';
        bmpBytes[1] = (byte)'M';
        bmpBytes[14] = 40;
        Array.Copy(BitConverter.GetBytes(bmpBytes.Length), 0, bmpBytes, 2, 4);
        Array.Copy(BitConverter.GetBytes(imageHeaderSize), 0, bmpBytes, 10, 4);
        Array.Copy(BitConverter.GetBytes(Width), 0, bmpBytes, 18, 4);
        Array.Copy(BitConverter.GetBytes(Height), 0, bmpBytes, 22, 4);
        Array.Copy(BitConverter.GetBytes(32), 0, bmpBytes, 28, 2);
        Array.Copy(BitConverter.GetBytes(ImageBytes.Length), 0, bmpBytes, 34, 4);
        Array.Copy(ImageBytes, 0, bmpBytes, imageHeaderSize, ImageBytes.Length);
        return bmpBytes;
    }

    public void Save(string filename)
    {
        byte[] bytes = GetBitmapBytes();
        File.WriteAllBytes(filename, bytes);
    }
}
```

## Generate Images from Scratch

The following code uses the bitmap class and color struct above to create test images

### Random Colors

```cs
RawBitmap bmp = new(400, 300);
Random rand = new();
for (int x = 0; x < bmp.Width; x++)
    for (int y = 0; y < bmp.Height; y++)
        bmp.SetPixel(x, y, RawColor.Random(rand));
bmp.Save("random-rgb.bmp");
```

![](SkiaSharp-random-rgb.jpg)

### Rainbow
```cs
RawBitmap bmp = new(400, 300);
Random rand = new();
for (int x = 0; x < bmp.Width; x++)
{
    for (int y = 0; y < bmp.Height; y++)
    {
        byte r = (byte)(255.0 * x / bmp.Width);
        byte g = (byte)(255.0 * y / bmp.Height);
        byte b = (byte)(255 - 255.0 * x / bmp.Width);
        RawColor color = new(r, g, b);
        bmp.SetPixel(x, y, color);
    }
}
bmp.Save("rainbow.bmp");
```

![](SkiaSharp-rainbow.jpg)

### Rectangles
```cs
RawBitmap bmp = new(400, 300);
Random rand = new();
for (int i = 0; i < 1000; i++)
{
    int rectX = rand.Next(bmp.Width);
    int rectY = rand.Next(bmp.Height);
    int rectWidth = rand.Next(50);
    int rectHeight = rand.Next(50);
    RawColor color = RawColor.Random(rand);

    for (int x = rectX; x < rectX + rectWidth; x++)
    {
        for (int y = rectY; y < rectY + rectHeight; y++)
        {
            if (x < 0 || x >= bmp.Width) continue;
            if (y < 0 || y >= bmp.Height) continue;
            bmp.SetPixel(x, y, color);
        }
    }
}
bmp.Save("rectangles.bmp");
```

![](SkiaSharp-rectangles.jpg)

## Interfacing Graphics Libraries

**The following code demonstrates how to load the bitmap byte arrays generated above into common graphics libraries and save the result as a JPEG file.** Although the bitmap byte array can be written directly to disk as a .bmp file, these third-party libraries are required to encode images in additional formats like JPEG.


### System.Drawing

```cs
using System.Drawing;

static void SaveBitmap(byte[] bytes, string filename = "demo.jpg")
{
    using MemoryStream ms = new(bytes);
    using Image img = Bitmap.FromStream(ms);
    img.Save(filename);
}
```

### SkiaSharp

```cs
using SkiaSharp;

static void SaveBitmap(byte[] bytes, string filename = "demo.jpg")
{
    using SKBitmap bmp = SKBitmap.Decode(bytes);
    using SKFileWStream fs = new(filename);
    bmp.Encode(fs, SKEncodedImageFormat.Jpeg, quality: 95);
}
```

### ImageSharp

```cs
using SixLabors.ImageSharp;
using SixLabors.ImageSharp.Formats.Jpeg;

static void SaveBitmap(byte[] bytes, string filename = "demo.jpg")
{
    using Image img = Image.Load(bytes);
    JpegEncoder encoder = new() { Quality = 95 };
    img.Save(filename, encoder);
}
```

# Resources

* [C# Data Visualization](https://swharden.com/csdv/) - Resources for visualizing data using C# and the .NET platform

* [SkiaSharp: `SKColor.cs`](https://github.com/mono/SkiaSharp/blob/main/binding/Binding/SKColor.cs)

* [System.Drawing: `Color.cs`](https://github.com/dotnet/runtime/blob/main/src/libraries/System.Drawing.Primitives/src/System/Drawing/Color.cs)

* [Maui.Graphics: `Color.cs`](https://github.com/dotnet/maui/blob/main/src/Graphics/src/Graphics/Color.cs)
October 16th, 2022

Experiments in PSK-31 Synthesis

PSK-31 is a narrow-bandwidth digital mode which encodes text as an audio tone that varies phase at a known rate. To learn more about this digital mode and solve a challenging programming problem, I'm going to write a PSK-31 encoder and decoder from scratch using the C# programming language. All code created for this project is open-source, available from my PSK Experiments GitHub Repository, and released under the permissive MIT license. This page documents my progress and notes things I learn along the way.

Encoding Bits as Phase Shifts

  • PSK31 messages have a continuous carrier tone

  • Symbols are represented by "symbols", each 1/31.25 seconds long

  • If a symbol changes phase from its previous symbol it is a 0, otherwise it is a 1

Amplitude Modulation Silences Phase Transitions

Although a continuous phase-shifting tone of constant amplitude can successfully transmit PSK31 data, the abrupt phase transitions will cause splatter. If you transmit this you will be heard, but those trying to communicate using adjacent frequencies will be highly disappointed.

Hard phase transitions (splatter) Soft phase transitions (cleaner)

To reduce spectral artifacts that result from abruptly changing phase, phase transitions are silenced by shaping the waveform envelope as a sine wave so it is silent at the transition. This way the maximum rate of phase shifts is a sine wave with a period of half the baud rate. This is why the opening of a PSK31 message (a series of logical 0 bits) sounds like two tones: It's the carrier sine wave with an envelope shaped like a sine wave with a period of 31.25/2 Hz. These two tones separated by approximately 15 Hz are visible in the spectrogram (waterfall).

Encoding Text as Bits

Unlike ASCII (8 bits per character) and RTTY (5 bits per character), BPSK uses Varicode (1-10 bits per character) to encode text. Consecutive zeros 00 separate characters, so character codes must not contain 00 and they must start ane end with a 1 bit. Messages are flanked by a preamble (repeated 0 bits) and a postamble (repeated 1 bits).

NUL 1010101011
SOH 1011011011
STX 1011101101
ETX 1101110111
EOT 1011101011
ENQ 1101011111
ACK 1011101111
BEL 1011111101
BS  1011111111
HT  11101111
LF  11101
VT  1101101111
FF  1011011101
CR  11111
SO  1101110101
SI  1110101011
DLE 1011110111
DC1 1011110101
DC2 1110101101
DC3 1110101111
DC4 1101011011
NAK 1101101011
SYN 1101101101
ETB 1101010111
CAN 1101111011
EM  1101111101
SUB 1110110111
ESC 1101010101
FS  1101011101
GS  1110111011
RS  1011111011
US  1101111111
SP  1
!   111111111
"   101011111
#   111110101
$   111011011
%   1011010101
&   1010111011
'   101111111
(   11111011
)   11110111
*   101101111
+   111011111
,   1110101
-   110101
.   1010111
/   110101111
0   10110111
1   10111101
2   11101101
3   11111111
4   101110111
5   101011011
6   101101011
7   110101101
8   110101011
9   110110111
:   11110101
;   110111101
<   111101101
=   1010101
>   111010111
?   1010101111
@   1010111101
A   1111101
B   11101011
C   10101101
D   10110101
E   1110111
F   11011011
G   11111101
H   101010101
I   1111111
J   111111101
K   101111101
L   11010111
M   10111011
N   11011101
O   10101011
P   11010101
Q   111011101
R   10101111
S   1101111
T   1101101
U   101010111
V   110110101
X   101011101
Y   101110101
Z   101111011
[   1010101101
\   111110111
]   111101111
^   111111011
_   1010111111
.   101101101
/   1011011111
a   1011
b   1011111
c   101111
d   101101
e   11
f   111101
g   1011011
h   101011
i   1101
j   111101011
k   10111111
l   11011
m   111011
n   1111
o   111
p   111111
q   110111111
r   10101
s   10111
t   101
u   110111
v   1111011
w   1101011
x   11011111
y   1011101
z   111010101
{   1010110111
|   110111011
}   1010110101
~   1011010111
DEL 1110110101

How to Generate a PSK Waveform

Now that we've covered the major steps of PSK31 message composition and modulation, let's go through the steps ot generate a PSK31 message in code.

Step 1: Convert a Message to Varicode

Here's the gist of how I store my varicode table in code. Note that the struct has an additional Description field which is useful for decoding and debugging.

public struct VaricodeSymbol
{
    public readonly string Symbol;
    public string BitString;
    public int[] Bits;
    public readonly string Description;

    public VaricodeSymbol(string symbol, string bitString, string? description = null)
    {
        Symbol = symbol;
        BitString = bitString;
        Bits = bitString.ToCharArray().Select(x => x == '1' ? 1 : 0).ToArray();
        Description = description ?? string.Empty;
    }
}
static VaricodeSymbol[] GetAllSymbols() => new VaricodeSymbol[]
{
    new("NUL", "1010101011", "Null character"),
    new("LF", "11101", "Line feed"),
    new("CR", "11111", "Carriage return"),
    new("SP", "1", "Space"),
    new("a", "1011"),
    new("b", "1011111"),
    new("c", "101111"),
    // etc...
};

I won't show how I do the message-to-varicode lookup, but it's trivial. Here's the final function I use to generate Varicode bits from a string:

static int[] GetVaricodeBits(string message)
{
    List<int> bits = new();

    // add a preamble of repeated zeros
    for (int i=0; i<20; i++)
        bits.Add(0);

    // encode each character of a message
    foreach (char character in message)
    {
        VaricodeSymbol symbol = Lookup(character);
        bits.AddRange(symbol.Bits);
        bits.AddRange(CharacterSeparator);
    }

    // add a postamble of repeated ones
    for (int i=0; i<20; i++)
        bits.Add(1);

    return bits.ToArray();
}

Step 2: Determine Phase Shifts

Now that we have our Varicode bits, we need to generate an array to indicate phase transitions. A transition occurs every time a bit changes value form the previous bit. This code returns phase as an array of double given the bits from a Varicode message.

public static double[] GetPhaseShifts(int[] bits, double phase1 = 0, double phase2 = Math.PI)
{
    double[] phases = new double[bits.Length];
    for (int i = 0; i < bits.Length; i++)
    {
        double previousPhase = i > 0 ? phases[i - 1] : phase1;
        double oppositePhase = previousPhase == phase1 ? phase2 : phase1;
        phases[i] = bits[i] == 1 ? previousPhase : oppositePhase;
    }
    return phases;
}

Step 3: Generate the Waveform

These constants will be used to define the shape of the waveform:

public const int SampleRate = 8000;
public const double Frequency = 1000;
public const double BaudRate = 31.25;

This minimal code generates a decipherable PSK-31 message, but it does not silence the phase transitions so it produces a lot of splatter. This function must be refined to shape the waveform such that phase transitions are silenced.

public double[] GetWaveformBPSK(double[] phases)
{
    int totalSamples = (int)(phases.Length * SampleRate / BaudRate);
    double[] wave = new double[totalSamples];
    for (int i = 0; i < wave.Length; i++)
    {
        double time = (double)i / SampleRate;
        int frame = (int)(time * BaudRate);
        double phaseShift = phases[frame];
        wave[i] = Math.Cos(2 * Math.PI * Frequency * time + phaseShift);
    }
    return wave;
}

Step 4: Generate the Waveform with Amplitude Modulation

This is the same function as above, but with extra logic for amplitude-modulating the waveform in the shape of a sine wave to silence phase transitions.

public double[] GetWaveformBPSK(double[] phases)
{
    int baudSamples = (int)(SampleRate / BaudRate);
    double samplesPerBit = SampleRate / BaudRate;
    int totalSamples = (int)(phases.Length * SampleRate / BaudRate);
    double[] wave = new double[totalSamples];

    // create the amplitude envelope sized for a single bit
    double[] envelope = new double[(int)samplesPerBit];
    for (int i = 0; i < envelope.Length; i++)
        envelope[i] = Math.Sin((i + .5) * Math.PI / envelope.Length);

    for (int i = 0; i < wave.Length; i++)
    {
        // phase modulated carrier
        double time = (double)i / SampleRate;
        int frame = (int)(time * BaudRate);
        double phaseShift = phases[frame];
        wave[i] = Math.Cos(2 * Math.PI * Frequency * time + phaseShift);

        // envelope at phase transitions
        int firstSample = (int)(frame * SampleRate / BaudRate);
        int distanceFromFrameStart = i - firstSample;
        int distanceFromFrameEnd = baudSamples - distanceFromFrameStart + 1;
        bool isFirstHalfOfFrame = distanceFromFrameStart < distanceFromFrameEnd;
        bool samePhaseAsLast = frame == 0 ? false : phases[frame - 1] == phases[frame];
        bool samePhaseAsNext = frame == phases.Length - 1 ? false : phases[frame + 1] == phases[frame];
        bool rampUp = isFirstHalfOfFrame && !samePhaseAsLast;
        bool rampDown = !isFirstHalfOfFrame && !samePhaseAsNext;

        if (rampUp)
            wave[i] *= envelope[distanceFromFrameStart];

        if (rampDown)
            wave[i] *= envelope[distanceFromFrameEnd];
    }

    return wave;
}

PSK31 Encoder Program

I wrapped the functionality above in a Windows Forms GUI that allows the user to type a message, specify frequency, baud rate, and whether or not to refine the envelope to reduce splatter, then either play or save the result. An interactive ScottPlot Chart allows the user to inspect the waveform.

Sample PSK-31 Transmissions

These audio files encode the text The Quick Brown Fox Jumped Over The Lazy Dog 1234567890 Times! in 1kHz BPSK at various baud rates.

Non-Standard Baud Rates

Let's see what PSK-3 sounds like. This mode encodes data at a rate of 3 bits per second. Note that Varicode characters may require up to ten bits, so this is pretty slow. On the other hand the side tones are closer to the carrier and the total bandwidth is much smaller. The message here has been shortened to just my callsign, AJ4VD.

Encode PSK-31 In Your Browser

After implementing the C# encoder described above I created a JavaScript version (as per Atwood's Law).

Decoding PSK-31

Considering all the steps for encoding PSK-31 transmissions are already described on this page, it doesn't require too much additional effort to create a decoder using basic software techniques. Once symbol phases are detected it's easy to work backwards: detect phase transitions (logical 0s) or repeats (logical 1s), treat consecutive zeros as a character separator, then look-up characters according to the Varicode table. The tricky bit is analyzing the source audio to generate the array of phase offsets.

The simplest way to decode PSK-31 transmissions leans on the fact that we already know the baud rate: 31.25 symbols per second, or one symbol every 256 samples at 8kHz sample rate. The audio signal can be segregated into many 256-sample bins, and processed by FFT. Once the center frequency is determined, the FFT power at this frequency can be calculated for each bin. Signal offset can be adjusted to minimize the imaginary component of the FFTs at the carrier frequency, then the real component will be strongly positive or negative, allowing phase transitions to be easily detected.

There are more advanced techniques to improve BPSK decoding, such as continuously adjusting frequency and phase alignment (synchronization). A Costas loop can help lock onto the carrier frequency while preserving its phase. Visit Simple BPSK31 Decoding with Python for an excellent demonstration of how to decode BPSK31 using these advanced techniques.

A crude C# implementation of a BPSK decoded is available on GitHub in the PSK Experiments repository

Encoding PSK-31 in Hardware

Since BPSK is just a carrier that applies periodic 180º phase-shifts, it's easy to generate in hardware by directly modulating the signal source. A good example of this is KA7OEI's PSK31 transmitter which feeds the output of an oscillator through an even or odd number of NAND gates (from a 74HC00) to produce two signals of opposite phase.

Quadrature Phase Shift Keying (QPSK)

Unlike the 0º and 180º phases of binary phase shift keying (BPSK), quadrature phase shift keying (QPSK) encodes extra data into each symbol by uses a larger number of phases. When QPSK-31 is used in amateur radio these extra bits aren't used to send messages faster but instead send them more reliably using convolutional coding and error correction. These additional features come at a cost (an extra 3 dB SNR is required), and in practice QPSK is not used as much by amateur radio operators.

QPSK encoding/decoding and convolutional encoding/decoding are outside the scope of this page, but excellent information exists on the Wikipedia: QPSK and in the US Naval Academy's EC314 Lesson 23: Digital Modulation document.

PSK-31 in 2022

After all that, it turns out PSK-31 isn't that popular anymore. These days it seems the FT-8 digital mode with WSJT-X software is orders of magnitude more popular ??

Resources

Markdown source code last modified on October 18th, 2022
---
Title: Experiments in PSK-31 Synthesis
Description: How to encode and decode PSK-31 messages using C#
Date: 2022-10-16 23:26PM EST
Tags: amateur radio, csharp
---

# Experiments in PSK-31 Synthesis

**PSK-31 is a narrow-bandwidth digital mode which encodes text as an audio tone that varies phase at a known rate.** To learn more about this digital mode and solve a challenging programming problem, I'm going to write a PSK-31 encoder and decoder from scratch using the C# programming language. All code created for this project is open-source, available from my [PSK Experiments GitHub Repository](https://github.com/swharden/psk-experiments), and released under the permissive MIT license. This page documents my progress and notes things I learn along the way.

<a href="psk-waterfall2.jpg"><img src="psk-waterfall2.jpg"></a>

## Encoding Bits as Phase Shifts

<!--
<img src="bpsk.png" class="w-75 d-block mx-auto my-3">
-->

* PSK31 messages have a continuous carrier tone

* Symbols are represented by "symbols", each 1/31.25 seconds long

* If a symbol changes phase from its previous symbol it is a `0`, otherwise it is a `1`

<a href="theory.png"><img src="theory.png" class="d-block mx-auto"></a>

## Amplitude Modulation Silences Phase Transitions

Although a continuous phase-shifting tone of constant amplitude can successfully transmit PSK31 data, the abrupt phase transitions will cause splatter. If you transmit this you will be heard, but those trying to communicate using adjacent frequencies will be highly disappointed.

<div class="text-center">

Hard phase transitions (splatter) | Soft phase transitions (cleaner)
---|---
<img src="splatter.png">|<img src="no-splatter.png">

</div>

To reduce spectral artifacts that result from abruptly changing phase, phase transitions are silenced by shaping the waveform envelope as a sine wave so it is silent at the transition. This way the maximum rate of phase shifts is a sine wave with a period of half the baud rate. This is why the opening of a PSK31 message (a series of logical `0` bits) sounds like two tones: It's the carrier sine wave with an envelope shaped like a sine wave with a period of 31.25/2 Hz. These two tones separated by approximately 15 Hz are visible in the spectrogram (waterfall).

![](modulation.png)

## Encoding Text as Bits

**Unlike ASCII (8 bits per character) and RTTY (5 bits per character), BPSK uses [Varicode](https://en.wikipedia.org/wiki/Varicode) (1-10 bits per character) to encode text.** Consecutive zeros `00` separate characters, so character codes must not contain `00` and they must start ane end with a `1` bit. Messages are flanked by a preamble (repeated `0` bits) and a postamble (repeated `1` bits).

<table cellpadding=30 style="white-space: nowrap;" class="m-0 p-0 mx-auto"><tr><td valign=top>
<TT>NUL 1010101011</TT>
<BR><TT>SOH 1011011011</TT>
<BR><TT>STX 1011101101</TT>
<BR><TT>ETX 1101110111</TT>
<BR><TT>EOT 1011101011</TT>
<BR><TT>ENQ 1101011111</TT>
<BR><TT>ACK 1011101111</TT>
<BR><TT>BEL 1011111101</TT>
<BR><TT>BS&nbsp; 1011111111</TT>
<BR><TT>HT&nbsp; 11101111</TT>
<BR><TT>LF&nbsp; 11101</TT>
<BR><TT>VT&nbsp; 1101101111</TT>
<BR><TT>FF&nbsp; 1011011101</TT>
<BR><TT>CR&nbsp; 11111</TT>
<BR><TT>SO&nbsp; 1101110101</TT>
<BR><TT>SI&nbsp; 1110101011</TT>
<BR><TT>DLE 1011110111</TT>
<BR><TT>DC1 1011110101</TT>
<BR><TT>DC2 1110101101</TT>
<BR><TT>DC3 1110101111</TT>
<BR><TT>DC4 1101011011</TT>
<BR><TT>NAK 1101101011</TT>
<BR><TT>SYN 1101101101</TT>
<BR><TT>ETB 1101010111</TT>
<BR><TT>CAN 1101111011</TT>
<BR><TT>EM&nbsp; 1101111101</TT>
<BR><TT>SUB 1110110111</TT>
<BR><TT>ESC 1101010101</TT>
<BR><TT>FS&nbsp; 1101011101</TT>
<BR><TT>GS&nbsp; 1110111011</TT>
<BR><TT>RS&nbsp; 1011111011</TT>
<BR><TT>US&nbsp; 1101111111</TT>
<BR><TT>SP&nbsp; 1</TT>
<BR><TT>!&nbsp;&nbsp; 111111111</TT>
<BR><TT>"&nbsp;&nbsp; 101011111</TT>
<BR><TT>#&nbsp;&nbsp; 111110101</TT>
<BR><TT>$&nbsp;&nbsp; 111011011</TT>
<BR><TT>%&nbsp;&nbsp; 1011010101</TT>
<BR><TT>&amp;&nbsp;&nbsp; 1010111011</TT>
<BR><TT>'&nbsp;&nbsp; 101111111</TT>
<BR><TT>(&nbsp;&nbsp; 11111011</TT>
<BR><TT>)&nbsp;&nbsp; 11110111</TT>
<BR><TT>*&nbsp;&nbsp; 101101111</TT>
</td><td valign=top width="33%" >
<TT>+&nbsp;&nbsp; 111011111</TT>
<BR><TT>,&nbsp;&nbsp; 1110101</TT>
<BR><TT>-&nbsp;&nbsp; 110101</TT>
<BR><TT>.&nbsp;&nbsp; 1010111</TT>
<BR><TT>/&nbsp;&nbsp; 110101111</TT>
<BR><TT>0&nbsp;&nbsp; 10110111</TT>
<BR><TT>1&nbsp;&nbsp; 10111101</TT>
<BR><TT>2&nbsp;&nbsp; 11101101</TT>
<BR><TT>3&nbsp;&nbsp; 11111111</TT>
<BR><TT>4&nbsp;&nbsp; 101110111</TT>
<BR><TT>5&nbsp;&nbsp; 101011011</TT>
<BR><TT>6&nbsp;&nbsp; 101101011</TT>
<BR><TT>7&nbsp;&nbsp; 110101101</TT>
<BR><TT>8&nbsp;&nbsp; 110101011</TT>
<BR><TT>9&nbsp;&nbsp; 110110111</TT>
<BR><TT>:&nbsp;&nbsp; 11110101</TT>
<BR><TT>;&nbsp;&nbsp; 110111101</TT>
<BR><TT>&lt;&nbsp;&nbsp; 111101101</TT>
<BR><TT>=&nbsp;&nbsp; 1010101</TT>
<BR><TT>>&nbsp;&nbsp; 111010111</TT>
<BR><TT>?&nbsp;&nbsp; 1010101111</TT>
<BR><TT>@&nbsp;&nbsp; 1010111101</TT>
<BR><TT>A&nbsp;&nbsp; 1111101</TT>
<BR><TT>B&nbsp;&nbsp; 11101011</TT>
<BR><TT>C&nbsp;&nbsp; 10101101</TT>
<BR><TT>D&nbsp;&nbsp; 10110101</TT>
<BR><TT>E&nbsp;&nbsp; 1110111</TT>
<BR><TT>F&nbsp;&nbsp; 11011011</TT>
<BR><TT>G&nbsp;&nbsp; 11111101</TT>
<BR><TT>H&nbsp;&nbsp; 101010101</TT>
<BR><TT>I&nbsp;&nbsp; 1111111</TT>
<BR><TT>J&nbsp;&nbsp; 111111101</TT>
<BR><TT>K&nbsp;&nbsp; 101111101</TT>
<BR><TT>L&nbsp;&nbsp; 11010111</TT>
<BR><TT>M&nbsp;&nbsp; 10111011</TT>
<BR><TT>N&nbsp;&nbsp; 11011101</TT>
<BR><TT>O&nbsp;&nbsp; 10101011</TT>
<BR><TT>P&nbsp;&nbsp; 11010101</TT>
<BR><TT>Q&nbsp;&nbsp; 111011101</TT>
<BR><TT>R&nbsp;&nbsp; 10101111</TT>
<BR><TT>S&nbsp;&nbsp; 1101111</TT>
<BR><TT>T&nbsp;&nbsp; 1101101</TT>
<BR><TT>U&nbsp;&nbsp; 101010111</TT>
</td><td valign=top width="33%" >
<TT>V&nbsp;&nbsp; 110110101</TT>
<BR><TT>X&nbsp;&nbsp; 101011101</TT>
<BR><TT>Y&nbsp;&nbsp; 101110101</TT>
<BR><TT>Z&nbsp;&nbsp; 101111011</TT>
<BR><TT>[&nbsp;&nbsp; 1010101101</TT>
<BR><TT>\&nbsp;&nbsp; 111110111</TT>
<BR><TT>]&nbsp;&nbsp; 111101111</TT>
<BR><TT>^&nbsp;&nbsp; 111111011</TT>
<BR><TT>_&nbsp;&nbsp; 1010111111</TT>
<BR><TT>.&nbsp;&nbsp; 101101101</TT>
<BR><TT>/&nbsp;&nbsp; 1011011111</TT>
<BR><TT>a&nbsp;&nbsp; 1011</TT>
<BR><TT>b&nbsp;&nbsp; 1011111</TT>
<BR><TT>c&nbsp;&nbsp; 101111</TT>
<BR><TT>d&nbsp;&nbsp; 101101</TT>
<BR><TT>e&nbsp;&nbsp; 11</TT>
<BR><TT>f&nbsp;&nbsp; 111101</TT>
<BR><TT>g&nbsp;&nbsp; 1011011</TT>
<BR><TT>h&nbsp;&nbsp; 101011</TT>
<BR><TT>i&nbsp;&nbsp; 1101</TT>
<BR><TT>j&nbsp;&nbsp; 111101011</TT>
<BR><TT>k&nbsp;&nbsp; 10111111</TT>
<BR><TT>l&nbsp;&nbsp; 11011</TT>
<BR><TT>m&nbsp;&nbsp; 111011</TT>
<BR><TT>n&nbsp;&nbsp; 1111</TT>
<BR><TT>o&nbsp;&nbsp; 111</TT>
<BR><TT>p&nbsp;&nbsp; 111111</TT>
<BR><TT>q&nbsp;&nbsp; 110111111</TT>
<BR><TT>r&nbsp;&nbsp; 10101</TT>
<BR><TT>s&nbsp;&nbsp; 10111</TT>
<BR><TT>t&nbsp;&nbsp; 101</TT>
<BR><TT>u&nbsp;&nbsp; 110111</TT>
<BR><TT>v&nbsp;&nbsp; 1111011</TT>
<BR><TT>w&nbsp;&nbsp; 1101011</TT>
<BR><TT>x&nbsp;&nbsp; 11011111</TT>
<BR><TT>y&nbsp;&nbsp; 1011101</TT>
<BR><TT>z&nbsp;&nbsp; 111010101</TT>
<BR><TT>{&nbsp;&nbsp; 1010110111</TT>
<BR><TT>|&nbsp;&nbsp; 110111011</TT>
<BR><TT>}&nbsp;&nbsp; 1010110101</TT>
<BR><TT>~&nbsp;&nbsp; 1011010111</TT>
<BR><TT>DEL 1110110101</TT>
</td></tr></table>

## How to Generate a PSK Waveform

Now that we've covered the major steps of PSK31 message composition and modulation, let's go through the steps ot generate a PSK31 message in code.

### Step 1: Convert a Message to Varicode

Here's the gist of how I store my varicode table in code. Note that the `struct` has an additional `Description` field which is useful for decoding and debugging.

```cs
public struct VaricodeSymbol
{
    public readonly string Symbol;
    public string BitString;
    public int[] Bits;
    public readonly string Description;

    public VaricodeSymbol(string symbol, string bitString, string? description = null)
    {
        Symbol = symbol;
        BitString = bitString;
        Bits = bitString.ToCharArray().Select(x => x == '1' ? 1 : 0).ToArray();
        Description = description ?? string.Empty;
    }
}
```

```cs
static VaricodeSymbol[] GetAllSymbols() => new VaricodeSymbol[]
{
    new("NUL", "1010101011", "Null character"),
    new("LF", "11101", "Line feed"),
    new("CR", "11111", "Carriage return"),
    new("SP", "1", "Space"),
    new("a", "1011"),
    new("b", "1011111"),
    new("c", "101111"),
    // etc...
};
```

I won't show how I do the message-to-varicode lookup, but it's trivial. Here's the final function I use to generate Varicode bits from a string:

```cs
static int[] GetVaricodeBits(string message)
{
    List<int> bits = new();

    // add a preamble of repeated zeros
    for (int i=0; i<20; i++)
        bits.Add(0);

    // encode each character of a message
    foreach (char character in message)
    {
        VaricodeSymbol symbol = Lookup(character);
        bits.AddRange(symbol.Bits);
        bits.AddRange(CharacterSeparator);
    }

    // add a postamble of repeated ones
    for (int i=0; i<20; i++)
        bits.Add(1);

    return bits.ToArray();
}
```

### Step 2: Determine Phase Shifts

Now that we have our Varicode bits, we need to generate an array to indicate phase transitions. A transition occurs every time a bit changes value form the previous bit. This code returns phase as an array of `double` given the bits from a Varicode message.

```cs
public static double[] GetPhaseShifts(int[] bits, double phase1 = 0, double phase2 = Math.PI)
{
    double[] phases = new double[bits.Length];
    for (int i = 0; i < bits.Length; i++)
    {
        double previousPhase = i > 0 ? phases[i - 1] : phase1;
        double oppositePhase = previousPhase == phase1 ? phase2 : phase1;
        phases[i] = bits[i] == 1 ? previousPhase : oppositePhase;
    }
    return phases;
}
```

### Step 3: Generate the Waveform

These constants will be used to define the shape of the waveform:

```cs
public const int SampleRate = 8000;
public const double Frequency = 1000;
public const double BaudRate = 31.25;
```

This minimal code generates a decipherable PSK-31 message, but it does not silence the phase transitions so it produces a lot of splatter. This function must be refined to shape the waveform such that phase transitions are silenced.

```cs
public double[] GetWaveformBPSK(double[] phases)
{
    int totalSamples = (int)(phases.Length * SampleRate / BaudRate);
    double[] wave = new double[totalSamples];
    for (int i = 0; i < wave.Length; i++)
    {
        double time = (double)i / SampleRate;
        int frame = (int)(time * BaudRate);
        double phaseShift = phases[frame];
        wave[i] = Math.Cos(2 * Math.PI * Frequency * time + phaseShift);
    }
    return wave;
}
```

### Step 4: Generate the Waveform with Amplitude Modulation

This is the same function as above, but with extra logic for amplitude-modulating the waveform in the shape of a sine wave to silence phase transitions.

```cs
public double[] GetWaveformBPSK(double[] phases)
{
    int baudSamples = (int)(SampleRate / BaudRate);
    double samplesPerBit = SampleRate / BaudRate;
    int totalSamples = (int)(phases.Length * SampleRate / BaudRate);
    double[] wave = new double[totalSamples];

    // create the amplitude envelope sized for a single bit
    double[] envelope = new double[(int)samplesPerBit];
    for (int i = 0; i < envelope.Length; i++)
        envelope[i] = Math.Sin((i + .5) * Math.PI / envelope.Length);

    for (int i = 0; i < wave.Length; i++)
    {
        // phase modulated carrier
        double time = (double)i / SampleRate;
        int frame = (int)(time * BaudRate);
        double phaseShift = phases[frame];
        wave[i] = Math.Cos(2 * Math.PI * Frequency * time + phaseShift);

        // envelope at phase transitions
        int firstSample = (int)(frame * SampleRate / BaudRate);
        int distanceFromFrameStart = i - firstSample;
        int distanceFromFrameEnd = baudSamples - distanceFromFrameStart + 1;
        bool isFirstHalfOfFrame = distanceFromFrameStart < distanceFromFrameEnd;
        bool samePhaseAsLast = frame == 0 ? false : phases[frame - 1] == phases[frame];
        bool samePhaseAsNext = frame == phases.Length - 1 ? false : phases[frame + 1] == phases[frame];
        bool rampUp = isFirstHalfOfFrame && !samePhaseAsLast;
        bool rampDown = !isFirstHalfOfFrame && !samePhaseAsNext;

        if (rampUp)
            wave[i] *= envelope[distanceFromFrameStart];

        if (rampDown)
            wave[i] *= envelope[distanceFromFrameEnd];
    }

    return wave;
}
```

## PSK31 Encoder Program

I wrapped the functionality above in a Windows Forms GUI that allows the user to type a message, specify frequency, baud rate, and whether or not to refine the envelope to reduce splatter, then either play or save the result. An interactive [ScottPlot Chart](https://scottplot.net) allows the user to inspect the waveform.

* **Download PSK31 Encoder: [PSK31-encoder.zip](PSK31-encoder.zip)**

* **PSK31 Encoder Source Code: [PSK Experiments on GitHub](https://github.com/swharden/psk-experiments)**

![](screenshot.png)

## Sample PSK-31 Transmissions

These audio files encode the text _The Quick Brown Fox Jumped Over The Lazy Dog 1234567890 Times!_ in 1kHz BPSK at various baud rates.

* PSK-31: [dog31.wav](dog31.wav)
* PSK-63: [dog63.wav](dog63.wav)
* PSK-125: [dog125.wav](dog125.wav)
* PSK-256: [dog256.wav](dog256.wav)

### Non-Standard Baud Rates

**Let's see what PSK-3 sounds like.** This mode encodes data at a rate of 3 bits per second. Note that Varicode characters may require up to ten bits, so this is pretty slow. On the other hand the side tones are closer to the carrier and the total bandwidth is much smaller. The message here has been shortened to just my callsign, AJ4VD.

* PSK-3: [psk3.wav](psk3.wav)

## Encode PSK-31 In Your Browser

After implementing the C# encoder described above I created a JavaScript version (as per <a href="https://en.wikipedia.org/wiki/Jeff_Atwood">Atwood's Law</a>).

* Try it on your phone or computer! [**Launch PskJS**](pskjs)

<a href="pskjs"><img src="pskjs.png" class="d-block mx-auto my-3"></a>

## Decoding PSK-31

Considering all the steps for _encoding_ PSK-31 transmissions are already described on this page, it doesn't require too much additional effort to create a _decoder_ using basic software techniques. Once symbol phases are detected it's easy to work backwards: detect phase transitions (logical 0s) or repeats (logical 1s), treat consecutive zeros as a character separator, then look-up characters according to the Varicode table. The tricky bit is analyzing the source audio to generate the array of phase offsets.

The simplest way to decode PSK-31 transmissions leans on the fact that we already know the baud rate: 31.25 symbols per second, or one symbol every 256 samples at 8kHz sample rate. The audio signal can be segregated into many 256-sample bins, and processed by FFT. Once the center frequency is determined, the FFT power at this frequency can be calculated for each bin. Signal offset can be adjusted to minimize the imaginary component of the FFTs at the carrier frequency, then the real component will be strongly positive or negative, allowing phase transitions to be easily detected.

<div class="row">
	<div class="col">
		<a href="psk31-receiver.html"><img src="py-fft.png" class="img-fluid"></a>
	</div>
	<div class="col">
		<a href="psk31-receiver.html"><img src="py-iq.png" class="img-fluid"></a>
	</div>
	<div class="col">
		<a href="psk31-receiver.html"><img src="py-eyediagram.png" class="img-fluid"></a>
	</div>
</div>

There are more advanced techniques to improve BPSK decoding, such as continuously adjusting frequency and phase alignment (synchronization). A [Costas loop](https://en.wikipedia.org/wiki/Costas_loop) can help lock onto the carrier frequency while preserving its phase. Visit [**Simple BPSK31 Decoding with Python**](psk31-receiver.html) for an excellent demonstration of how to decode BPSK31 using these advanced techniques.

A crude C# implementation of a BPSK decoded is available on GitHub in the [PSK Experiments](https://github.com/swharden/psk-experiments) repository

![](psk-decode.png)

## Encoding PSK-31 in Hardware

Since BPSK is just a carrier that applies periodic 180º phase-shifts, it's easy to generate in hardware by directly modulating the signal source. A good example of this is [KA7OEI's PSK31 transmitter](http://www.ka7oei.com/psk_bm_tx.html) which feeds the output of an oscillator through an even or odd number of NAND gates (from a [74HC00](https://www.mouser.com/datasheet/2/308/74HC00-105628.pdf)) to produce two signals of opposite phase.

![](pic-psk31.png)

## Quadrature Phase Shift Keying (QPSK)

**Unlike the 0º and 180º phases of binary phase shift keying (BPSK), quadrature phase shift keying (QPSK) encodes extra data into each symbol by uses a larger number of phases.** When QPSK-31 is used in amateur radio these extra bits aren't used to send messages faster but instead send them more reliably using convolutional coding and error correction. These additional features come at a cost (an extra 3 dB SNR is required), and in practice QPSK is not used as much by amateur radio operators.

QPSK encoding/decoding and convolutional encoding/decoding are outside the scope of this page, but excellent information exists on the [Wikipedia: QPSK](https://en.wikipedia.org/wiki/Phase-shift_keying#Quadrature_phase-shift_keying_(QPSK)) and in the US Naval Academy's [EC314 Lesson 23: Digital Modulation](https://www.usna.edu/ECE/ec312/Lessons/wireless/EC312_Lesson_23_Digital_Modulation_Course_Notes.pdf) document.

<a href="qpsk.png"><img src="qpsk.png" class="mx-auto d-block"></a>

## PSK-31 in 2022

After all that, it turns out PSK-31 isn't that popular anymore. These days it seems the [FT-8 digital mode](https://en.wikipedia.org/wiki/FT8) with [WSJT-X software](https://physics.princeton.edu/pulsar/k1jt/wsjtx.html) is orders of magnitude more popular ??

## Resources

* [PSK Experiments](https://github.com/swharden/psk-experiments) (GitHub) - Source code for the project shown on this page

* Software: [digipan](https://www.apkfollow.com/articles/2020/06/digipan.net.html) - A Freeware Program for PSK31 and PSK63

* Software: [fldigi](http://www.w1hkj.com/) - Supports PSK31 and other digital modes

* Software: [Digital Master 780](https://swharden.com/blog/2022-10-14-ham-radio-deluxe) - A Windows application that supports PSK31 and other digital modes bundled with Ham Radio Deluxe 5.0 (the last free version)

* Software: [WinPSK](https://www.moetronix.com/ae4jy/winpsk.htm) - open source PSK31 software for Windows

* Software: [PSKCore DLL](http://www.moetronix.com/ae4jy/pskcoredll.htm) - A Windows DLL that can be included in other software to add support for PSK31

* Software: [jacobwgillespie/psk31](https://github.com/jacobwgillespie/psk31) - Example PSK31 message generate using JavaScript

* [Digital Modulation](https://www.usna.edu/ECE/ec312/Lessons/wireless/EC312_Lesson_23_Digital_Modulation_Course_Notes.pdf) (US Naval Academy, EC314 Lesson 23) - A good description of quadrature PSK and higher order phase-shift encoding.

* [PSK-31 Specification](http://www.arrl.org/psk31-spec) (ARRL) - theory, varicode table, and convolutional code table.

* [PSK31 Description](http://aintel.bi.ehu.es/psk31.html) by G3PLX is the original / official description of the digital mode.

* [PSK31: A New Radio-Teletype Mode](http://www.arrl.org/files/file/Technology/tis/info/pdf/x9907003.pdf) (1999) by Peter Martinez, G3PLX

* [PSK31 The Easy Way](https://www.vic.wicen.org.au/wp-content/uploads/2012/05/psk31.pdf) (1999) by Alan Gibbs, VK6PG

* [Wikipedia: Varicode](https://en.wikipedia.org/wiki/Varicode) includes a table of all symbols

* [Wikipedia: QPSK](https://en.wikipedia.org/wiki/Phase-shift_keying#Quadrature_phase-shift_keying_(QPSK))

* [PSK31 Fundamentals](http://aintel.bi.ehu.es/psk31theory.html) and [PSK31 Setup](https://myplace.frontier.com/~nb6z/psk31.htm) by Peter Martinez, G3PLX

* [Varicode](http://math0.wvstateu.edu/~baker/cs240/info/varicode.html) - West Virginia State University CS240

* [Introduction to PSK31](https://sites.google.com/site/psk31matlabproject/introduction-to-psk) by engineering students at Walla Walla University

* [GNURadio PSK31 Decoder](https://sdradventure.wordpress.com/2011/10/15/gnuradio-psk31-decoder-part-1/) by VA7STH

* [Simple BPSK31](psk31-receiver.html) - a fantastic Jupyter notebook demonstrating BPSK decoding with Python

* [PySDR: Digital Modulation](https://pysdr.org/content/digital_modulation.html) - a summary of signal modulation types

* [A PIC-Based PSK31 exciter using a Balanced Modulator](http://www.ka7oei.com/psk_bm_tx.html) by Clint Turner, KA7OEI
June 23rd, 2022

Resample Time Series Data using Cubic Spline Interpolation

Cubic spline interpolation can be used to modify the sample rate of time series data. This page describes how I achieve signal resampling with spline interpolation in pure C# without any external dependencies. This technique can be used to:

  • Convert unevenly-sampled data to a series of values with a fixed sample rate

  • Convert time series data from one sample rate to another sample rate

  • Fill-in missing values from a collection of measurements

Simulating Data Samples

To simulate unevenly-sampled data I create a theoretical signal then sample it at 20 random time points.

// this function represents the signal being measured
static double f(double x) => Math.Sin(x * 10) + Math.Sin(x * 13f);

// randomly sample values from 20 time points
Random rand = new(123);
double[] sampleXs = Enumerable.Range(0, 20)
    .Select(x => rand.NextDouble())
    .OrderBy(x => x)
    .ToArray();
double[] sampleYs = sampleXs.Select(x => f(x)).ToArray();

Resample for Evenly-Spaced Data

I then generate an interpolated spline using my sampled data points as the input.

  • I can control the sample rate by defining the number of points generated in the output signal.

  • Full source code is at the bottom of this article.

(double[] xs, double[] ys) = Cubic.Interpolate1D(sampleXs, sampleYs, count: 50);

The generated points line-up perfectly with the sampled data.

There is slight deviation from the theoretical signal (and it's larger where there is more missing data) but this is an unsurprising result considering the original samples had large gaps of missing data.

Source Code

Interpolation.cs

public static class Interpolation
{
    public static (double[] xs, double[] ys) Interpolate1D(double[] xs, double[] ys, int count)
    {
        if (xs is null || ys is null || xs.Length != ys.Length)
            throw new ArgumentException($"{nameof(xs)} and {nameof(ys)} must have same length");

        int inputPointCount = xs.Length;
        double[] inputDistances = new double[inputPointCount];
        for (int i = 1; i < inputPointCount; i++)
            inputDistances[i] = inputDistances[i - 1] + xs[i] - xs[i - 1];

        double meanDistance = inputDistances.Last() / (count - 1);
        double[] evenDistances = Enumerable.Range(0, count).Select(x => x * meanDistance).ToArray();
        double[] xsOut = Interpolate(inputDistances, xs, evenDistances);
        double[] ysOut = Interpolate(inputDistances, ys, evenDistances);
        return (xsOut, ysOut);
    }

    private static double[] Interpolate(double[] xOrig, double[] yOrig, double[] xInterp)
    {
        (double[] a, double[] b) = FitMatrix(xOrig, yOrig);

        double[] yInterp = new double[xInterp.Length];
        for (int i = 0; i < yInterp.Length; i++)
        {
            int j;
            for (j = 0; j < xOrig.Length - 2; j++)
                if (xInterp[i] <= xOrig[j + 1])
                    break;

            double dx = xOrig[j + 1] - xOrig[j];
            double t = (xInterp[i] - xOrig[j]) / dx;
            double y = (1 - t) * yOrig[j] + t * yOrig[j + 1] +
                t * (1 - t) * (a[j] * (1 - t) + b[j] * t);
            yInterp[i] = y;
        }

        return yInterp;
    }

    private static (double[] a, double[] b) FitMatrix(double[] x, double[] y)
    {
        int n = x.Length;
        double[] a = new double[n - 1];
        double[] b = new double[n - 1];
        double[] r = new double[n];
        double[] A = new double[n];
        double[] B = new double[n];
        double[] C = new double[n];

        double dx1, dx2, dy1, dy2;

        dx1 = x[1] - x[0];
        C[0] = 1.0f / dx1;
        B[0] = 2.0f * C[0];
        r[0] = 3 * (y[1] - y[0]) / (dx1 * dx1);

        for (int i = 1; i < n - 1; i++)
        {
            dx1 = x[i] - x[i - 1];
            dx2 = x[i + 1] - x[i];
            A[i] = 1.0f / dx1;
            C[i] = 1.0f / dx2;
            B[i] = 2.0f * (A[i] + C[i]);
            dy1 = y[i] - y[i - 1];
            dy2 = y[i + 1] - y[i];
            r[i] = 3 * (dy1 / (dx1 * dx1) + dy2 / (dx2 * dx2));
        }

        dx1 = x[n - 1] - x[n - 2];
        dy1 = y[n - 1] - y[n - 2];
        A[n - 1] = 1.0f / dx1;
        B[n - 1] = 2.0f * A[n - 1];
        r[n - 1] = 3 * (dy1 / (dx1 * dx1));

        double[] cPrime = new double[n];
        cPrime[0] = C[0] / B[0];
        for (int i = 1; i < n; i++)
            cPrime[i] = C[i] / (B[i] - cPrime[i - 1] * A[i]);

        double[] dPrime = new double[n];
        dPrime[0] = r[0] / B[0];
        for (int i = 1; i < n; i++)
            dPrime[i] = (r[i] - dPrime[i - 1] * A[i]) / (B[i] - cPrime[i - 1] * A[i]);

        double[] k = new double[n];
        k[n - 1] = dPrime[n - 1];
        for (int i = n - 2; i >= 0; i--)
            k[i] = dPrime[i] - cPrime[i] * k[i + 1];

        for (int i = 1; i < n; i++)
        {
            dx1 = x[i] - x[i - 1];
            dy1 = y[i] - y[i - 1];
            a[i - 1] = k[i - 1] * dx1 - dy1;
            b[i - 1] = -k[i] * dx1 + dy1;
        }

        return (a, b);
    }
}

Program.cs

This is the source code I used to generate the figures on this page.

Plots were generated using ScottPlot.NET.

// this function represents the signal being measured
static double f(double x) => Math.Sin(x * 10) + Math.Sin(x * 13f);

// create points representing randomly sampled time points of a smooth curve
Random rand = new(123);
double[] sampleXs = Enumerable.Range(0, 20)
    .Select(x => rand.NextDouble())
    .OrderBy(x => x)
    .ToArray();
double[] sampleYs = sampleXs.Select(x => f(x)).ToArray();

// use 1D interpolation to create an evenly sampled curve from unevenly sampled data
(double[] xs, double[] ys) = Interpolation.Interpolate1D(sampleXs, sampleYs, count: 50);

var plt = new ScottPlot.Plot(600, 400);

double[] theoreticalXs = ScottPlot.DataGen.Range(xs.Min(), xs.Max(), .01);
double[] theoreticalYs = theoreticalXs.Select(x => f(x)).ToArray();
var perfectPlot = plt.AddScatterLines(theoreticalXs, theoreticalYs);
perfectPlot.Label = "theoretical signal";
perfectPlot.Color = plt.Palette.GetColor(2);
perfectPlot.LineStyle = ScottPlot.LineStyle.Dash;

var samplePlot = plt.AddScatterPoints(sampleXs, sampleYs);
samplePlot.Label = "sampled points";
samplePlot.Color = plt.Palette.GetColor(0);
samplePlot.MarkerSize = 10;
samplePlot.MarkerShape = ScottPlot.MarkerShape.openCircle;
samplePlot.MarkerLineWidth = 2;

var smoothPlot = plt.AddScatter(xs, ys);
smoothPlot.Label = "interpolated points";
smoothPlot.Color = plt.Palette.GetColor(3);
smoothPlot.MarkerShape = ScottPlot.MarkerShape.filledCircle;

plt.Legend();
plt.SaveFig("output.png");

2D and 3D Spline Interpolation

The interpolation method described above only considered the horizontal axis when generating evenly-spaced time points (1D interpolation). For information and code examples regarding 2D and 3D cubic spline interpolation, see my previous blog post: Spline Interpolation with C#

Resources

Markdown source code last modified on June 24th, 2022
---
Title: Resample Time Series Data using Cubic Spline Interpolation
Description: This page describes how I achieve signal resampling with spline interpolation in pure C# without any external dependencies.
Date: 2022-06-23 21:15PM EST
Tags: csharp, graphics
---

# Resample Time Series Data using Cubic Spline Interpolation

**Cubic spline interpolation can be used to modify the sample rate of time series data.** This page describes how I achieve signal resampling with spline interpolation in pure C# without any external dependencies. This technique can be used to:

* Convert unevenly-sampled data to a series of values with a fixed sample rate

* Convert time series data from one sample rate to another sample rate

* Fill-in missing values from a collection of measurements

### Simulating Data Samples

To simulate unevenly-sampled data I create a theoretical signal then sample it at 20 random time points.

```cs
// this function represents the signal being measured
static double f(double x) => Math.Sin(x * 10) + Math.Sin(x * 13f);

// randomly sample values from 20 time points
Random rand = new(123);
double[] sampleXs = Enumerable.Range(0, 20)
    .Select(x => rand.NextDouble())
    .OrderBy(x => x)
    .ToArray();
double[] sampleYs = sampleXs.Select(x => f(x)).ToArray();
```

<img src="1-samples-only.png" class="mx-auto d-block mb-5">

### Resample for Evenly-Spaced Data

I then generate an interpolated spline using my sampled data points as the input. 

* I can control the sample rate by defining the number of points generated in the output signal. 

* Full source code is at the bottom of this article.

```cs
(double[] xs, double[] ys) = Cubic.Interpolate1D(sampleXs, sampleYs, count: 50);
```

<img src="2-resample-only.png" class="mx-auto d-block mb-5">

The generated points line-up perfectly with the sampled data.

<img src="2-resample.png" class="mx-auto d-block mb-5">

There is slight deviation from the theoretical signal (and it's larger where there is more missing data) but this is an unsurprising result considering the original samples had large gaps of missing data.

<img src="3-comparison.png" class="mx-auto d-block mb-5">

## Source Code

### Interpolation.cs

```cs
public static class Interpolation
{
    public static (double[] xs, double[] ys) Interpolate1D(double[] xs, double[] ys, int count)
    {
        if (xs is null || ys is null || xs.Length != ys.Length)
            throw new ArgumentException($"{nameof(xs)} and {nameof(ys)} must have same length");

        int inputPointCount = xs.Length;
        double[] inputDistances = new double[inputPointCount];
        for (int i = 1; i < inputPointCount; i++)
            inputDistances[i] = inputDistances[i - 1] + xs[i] - xs[i - 1];

        double meanDistance = inputDistances.Last() / (count - 1);
        double[] evenDistances = Enumerable.Range(0, count).Select(x => x * meanDistance).ToArray();
        double[] xsOut = Interpolate(inputDistances, xs, evenDistances);
        double[] ysOut = Interpolate(inputDistances, ys, evenDistances);
        return (xsOut, ysOut);
    }

    private static double[] Interpolate(double[] xOrig, double[] yOrig, double[] xInterp)
    {
        (double[] a, double[] b) = FitMatrix(xOrig, yOrig);

        double[] yInterp = new double[xInterp.Length];
        for (int i = 0; i < yInterp.Length; i++)
        {
            int j;
            for (j = 0; j < xOrig.Length - 2; j++)
                if (xInterp[i] <= xOrig[j + 1])
                    break;

            double dx = xOrig[j + 1] - xOrig[j];
            double t = (xInterp[i] - xOrig[j]) / dx;
            double y = (1 - t) * yOrig[j] + t * yOrig[j + 1] +
                t * (1 - t) * (a[j] * (1 - t) + b[j] * t);
            yInterp[i] = y;
        }

        return yInterp;
    }

    private static (double[] a, double[] b) FitMatrix(double[] x, double[] y)
    {
        int n = x.Length;
        double[] a = new double[n - 1];
        double[] b = new double[n - 1];
        double[] r = new double[n];
        double[] A = new double[n];
        double[] B = new double[n];
        double[] C = new double[n];

        double dx1, dx2, dy1, dy2;

        dx1 = x[1] - x[0];
        C[0] = 1.0f / dx1;
        B[0] = 2.0f * C[0];
        r[0] = 3 * (y[1] - y[0]) / (dx1 * dx1);

        for (int i = 1; i < n - 1; i++)
        {
            dx1 = x[i] - x[i - 1];
            dx2 = x[i + 1] - x[i];
            A[i] = 1.0f / dx1;
            C[i] = 1.0f / dx2;
            B[i] = 2.0f * (A[i] + C[i]);
            dy1 = y[i] - y[i - 1];
            dy2 = y[i + 1] - y[i];
            r[i] = 3 * (dy1 / (dx1 * dx1) + dy2 / (dx2 * dx2));
        }

        dx1 = x[n - 1] - x[n - 2];
        dy1 = y[n - 1] - y[n - 2];
        A[n - 1] = 1.0f / dx1;
        B[n - 1] = 2.0f * A[n - 1];
        r[n - 1] = 3 * (dy1 / (dx1 * dx1));

        double[] cPrime = new double[n];
        cPrime[0] = C[0] / B[0];
        for (int i = 1; i < n; i++)
            cPrime[i] = C[i] / (B[i] - cPrime[i - 1] * A[i]);

        double[] dPrime = new double[n];
        dPrime[0] = r[0] / B[0];
        for (int i = 1; i < n; i++)
            dPrime[i] = (r[i] - dPrime[i - 1] * A[i]) / (B[i] - cPrime[i - 1] * A[i]);

        double[] k = new double[n];
        k[n - 1] = dPrime[n - 1];
        for (int i = n - 2; i >= 0; i--)
            k[i] = dPrime[i] - cPrime[i] * k[i + 1];

        for (int i = 1; i < n; i++)
        {
            dx1 = x[i] - x[i - 1];
            dy1 = y[i] - y[i - 1];
            a[i - 1] = k[i - 1] * dx1 - dy1;
            b[i - 1] = -k[i] * dx1 + dy1;
        }

        return (a, b);
    }
}
```

### Program.cs

This is the source code I used to generate the figures on this page. 

Plots were generated using [ScottPlot.NET](https://scottplot.net).

```cs
// this function represents the signal being measured
static double f(double x) => Math.Sin(x * 10) + Math.Sin(x * 13f);

// create points representing randomly sampled time points of a smooth curve
Random rand = new(123);
double[] sampleXs = Enumerable.Range(0, 20)
    .Select(x => rand.NextDouble())
    .OrderBy(x => x)
    .ToArray();
double[] sampleYs = sampleXs.Select(x => f(x)).ToArray();

// use 1D interpolation to create an evenly sampled curve from unevenly sampled data
(double[] xs, double[] ys) = Interpolation.Interpolate1D(sampleXs, sampleYs, count: 50);

var plt = new ScottPlot.Plot(600, 400);

double[] theoreticalXs = ScottPlot.DataGen.Range(xs.Min(), xs.Max(), .01);
double[] theoreticalYs = theoreticalXs.Select(x => f(x)).ToArray();
var perfectPlot = plt.AddScatterLines(theoreticalXs, theoreticalYs);
perfectPlot.Label = "theoretical signal";
perfectPlot.Color = plt.Palette.GetColor(2);
perfectPlot.LineStyle = ScottPlot.LineStyle.Dash;

var samplePlot = plt.AddScatterPoints(sampleXs, sampleYs);
samplePlot.Label = "sampled points";
samplePlot.Color = plt.Palette.GetColor(0);
samplePlot.MarkerSize = 10;
samplePlot.MarkerShape = ScottPlot.MarkerShape.openCircle;
samplePlot.MarkerLineWidth = 2;

var smoothPlot = plt.AddScatter(xs, ys);
smoothPlot.Label = "interpolated points";
smoothPlot.Color = plt.Palette.GetColor(3);
smoothPlot.MarkerShape = ScottPlot.MarkerShape.filledCircle;

plt.Legend();
plt.SaveFig("output.png");
```

## 2D and 3D Spline Interpolation

**The interpolation method described above only considered the horizontal axis** when generating evenly-spaced time points (1D interpolation). For information and code examples regarding 2D and 3D cubic spline interpolation, see my previous blog post: [Spline Interpolation with C#](https://swharden.com/blog/2022-01-22-spline-interpolation/) 

<a href="https://swharden.com/blog/2022-01-22-spline-interpolation/"><img src="https://swharden.com/blog/2022-01-22-spline-interpolation/screenshot.gif" class="mx-auto d-block mb-5"></a>

## Resources

* [2D and 3D Spline Interpolation with C#](https://swharden.com/blog/2022-01-22-spline-interpolation/) - Blog post from Jan, 2022

* [Spline Interpolation with ScottPlot](https://scottplot.net/cookbook/4.1/category/misc/#spline-interpolation) - Demonstrates additional types of interpolation: Bezier, Catmull-Rom, Chaikin, Cubic, etc. The project is open source under a MIT license.

* [Programmer's guide to polynomials and splines](https://wordsandbuttons.online/programmers_guide_to_polynomials_and_splines.html)
May 29th, 2022

Show A Progress Bar as your Blazor App Loads

Today I created a Blazor WebAssembly app that shows a progress bar while the page loads. This is especially useful for users on slow connections because Blazor apps typically require several megabytes of DLL and DAT files to be downloaded before meaningful content appears on the page.

Live Demo: LJPcalc (source code)

Step 1: Add a progress bar

Edit index.html and identify your app's main div:

<div id="app">Loading...</div>

Add a progress bar inside it:

<div id="app">
    <h2>Loading...</h2>
    <div class="progress mt-2" style="height: 2em;">
        <div id="progressbar" class="progress-bar progress-bar-striped progress-bar-animated"
            style="width: 10%; background-color: #204066;"></div>
    </div>
    <div>
        <span id="progressLabel" class="text-muted">Downloading file list</span>
    </div>
</div>

See Bootstrap's progressbar page for extensive customization and animation options and best practices when working with progress indicators. Also ensure the version of Bootstrap in your Blazor app is consistent with the documentation/HTML you are referencing.

Step 2: Disable Blazor AutoStart

Edit index.html and identify where your app loads Blazor resources:

<script src="_framework/blazor.webassembly.js"></script>

Update that script so it does not download automatically:

<script src="_framework/blazor.webassembly.js" autostart="false"></script>

Step 3: Create a Blazor Startup Script

Add a script to the bottom of the page to start Blazor manually, identifying all the resources needed and incrementally downloading them while updating the progressbar along the way.

<script>
    function StartBlazor() {
        let loadedCount = 0;
        const resourcesToLoad = [];
        Blazor.start({
            loadBootResource:
                function (type, filename, defaultUri, integrity) {
                    if (type == "dotnetjs")
                        return defaultUri;

                    const fetchResources = fetch(defaultUri, {
                        cache: 'no-cache',
                        integrity: integrity,
                        headers: { 'MyCustomHeader': 'My custom value' }
                    });

                    resourcesToLoad.push(fetchResources);

                    fetchResources.then((r) => {
                        loadedCount += 1;
                        if (filename == "blazor.boot.json")
                            return;
                        const totalCount = resourcesToLoad.length;
                        const percentLoaded = 10 + parseInt((loadedCount * 90.0) / totalCount);
                        const progressbar = document.getElementById('progressbar');
                        progressbar.style.width = percentLoaded + '%';
                        const progressLabel = document.getElementById('progressLabel');
                        progressLabel.innerText = `Downloading ${loadedCount}/${totalCount}: ${filename}`;
                    });

                    return fetchResources;
                }
        });
    }

    StartBlazor();
</script>

Resources

Markdown source code last modified on May 29th, 2022
---
title: Show A Progress Bar as your Blazor App Loads
description: How to add a progress bar to your client-side Blazor WebAssembly app to indicate page load progress.
date: 2022-05-29 16:05:00 EST
tags: blazor, csharp
---

# Show A Progress Bar as your Blazor App Loads

**Today I created a Blazor WebAssembly app that shows a progress bar while the page loads.** This is especially useful for users on slow connections because Blazor apps typically require several megabytes of DLL and DAT files to be downloaded before meaningful content appears on the page.

Live Demo: [LJPcalc](https://swharden.com/LJPcalc/) ([source code](https://github.com/swharden/LJPcalc))

<img src="blazor-load-progress-v2.gif" class="mx-auto d-block border shadow my-5">

## Step 1: Add a progress bar

Edit index.html and identify your app's main div:

```html
<div id="app">Loading...</div>
```

Add a progress bar inside it:
```html
<div id="app">
	<h2>Loading...</h2>
	<div class="progress mt-2" style="height: 2em;">
		<div id="progressbar" class="progress-bar progress-bar-striped progress-bar-animated"
			style="width: 10%; background-color: #204066;"></div>
	</div>
	<div>
		<span id="progressLabel" class="text-muted">Downloading file list</span>
	</div>
</div>
```

See [Bootstrap's progressbar page](https://getbootstrap.com/docs/5.2/components/progress/) for extensive customization and animation options and best practices when working with progress indicators. Also ensure the version of Bootstrap in your Blazor app is consistent with the documentation/HTML you are referencing.

## Step 2: Disable Blazor AutoStart

Edit index.html and identify where your app loads Blazor resources:

```html
<script src="_framework/blazor.webassembly.js"></script>
```

Update that script so it does _not_ download automatically:
```html
<script src="_framework/blazor.webassembly.js" autostart="false"></script>
```

## Step 3: Create a Blazor Startup Script

Add a script to the bottom of the page to start Blazor manually, identifying all the resources needed and incrementally downloading them while updating the progressbar along the way.

```html
<script>
	function StartBlazor() {
		let loadedCount = 0;
		const resourcesToLoad = [];
		Blazor.start({
			loadBootResource:
				function (type, filename, defaultUri, integrity) {
					if (type == "dotnetjs")
						return defaultUri;

					const fetchResources = fetch(defaultUri, {
						cache: 'no-cache',
						integrity: integrity,
						headers: { 'MyCustomHeader': 'My custom value' }
					});

					resourcesToLoad.push(fetchResources);

					fetchResources.then((r) => {
						loadedCount += 1;
						if (filename == "blazor.boot.json")
							return;
						const totalCount = resourcesToLoad.length;
						const percentLoaded = 10 + parseInt((loadedCount * 90.0) / totalCount);
						const progressbar = document.getElementById('progressbar');
						progressbar.style.width = percentLoaded + '%';
						const progressLabel = document.getElementById('progressLabel');
						progressLabel.innerText = `Downloading ${loadedCount}/${totalCount}: ${filename}`;
					});

					return fetchResources;
				}
		});
	}

	StartBlazor();
</script>
```

## Resources

* [ASP.NET Core Blazor startup](https://docs.microsoft.com/en-us/aspnet/core/blazor/fundamentals/startup) describes the `Blazor.start()` process and `loadBootResource()`.

* Code here inspired by [@EdmondShtogu's comment](https://github.com/dotnet/aspnetcore/issues/25165#issuecomment-781683925) on [dotnet/aspnetcore #25165](https://github.com/dotnet/aspnetcore/issues/25165) from Feb 18, 2021
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