Transmitting BPSK symbols (Part 2/5) 🐀

In the last post, we worked through what IQ is, and different formats that it may be sent or received in. Let’s take that and move on to Transmitting BPSK using IQ data!

When we transmit and receive information through RF using an SDR, data is traditionally encoded into a stream of symbols which are then used by a program to modulate the IQ stream, and sent over the airwaves.

PACKRAT uses BPSK to encode Symbols through RF. BPSK is the act of modulating the phase of a sine wave to carry information. The transmitted wave swaps between two states in order to convey a 0 or a 1. Our symbols modulate the transmitted sine wave’s phase, so that it moves between in-phase with the SDR’s transmitter and 180 degrees (or π radians) out of phase with the SDR’s transmitter.

The difference between a “Bit” and a “Symbol” in PACKRAT is not incredibly meaningful, and I’ll often find myself slipping up when talking about them. I’ve done my best to try and use the right word at the right stage, but it’s not as obvious where the line between bit and symbol is – at least not as obvious as it would be with QPSK or QAM. The biggest difference is that there are three meaningful states for PACKRAT over BPSK - a 1 (for “In phase”), -1 (for “180 degrees out of phase”) and 0 (for “no carrier”). For my implementation, a stream of all zeros will not transmit data over the airwaves, a stream of all 1s will transmit all “1” bits over the airwaves, and a stream of all -1s will transmit all “0” bits over the airwaves.

We’re not going to cover turning a byte (or bit) into a symbol yet – I’m going to write more about that in a later section. So for now, let’s just worry about symbols in, and symbols out.

Transmitting a Sine wave at 0Hz

If we go back to thinking about IQ data as a precisely timed measurements of energy over time at some particular specific frequency, we can consider what a sine wave will look like in IQ. Before we dive into antennas and RF, let’s go to something a bit more visual.

For the first example, you can see an example of a camera who’s frame rate (or Sampling Rate!) matches the exact number of rotations per second (or Frequency!) of the propeller and it appears to stand exactly still. Every time the Camera takes a frame, it’s catching the propeller in the exact same place in space, even though it’s made a complete rotation.

The second example is very similar, it’s a light strobing (in this case, our sampling rate, since the darkness is ignored by our brains) at the same rate (frequency) as water dropping from a faucet – and the video creator is even nice enough to change the sampling frequency to have the droplets move both forward and backward (positive and negative frequency) in comparison to the faucet.

IQ works the same way. If we catch something in perfect frequency alignment with our radio, we’ll wind up with readings that are the same for the entire stream of data. This means we can transmit a sine wave by setting all of the IQ samples in our buffer to 1+0i, which will transmit a pure sine wave at exactly the center frequency of the radio.

    var sine []complex{}
    for i := range sine {
        sine[i] = complex(1.0, 0.0)
    }

Alternatively, we can transmit a Sine wave (but with the opposite phase) by flipping the real value from 1 to -1. The same Sine wave is transmitted on the same Frequency, except when the wave goes high in the example above, the wave will go low in the example below.

    var sine []complex{}
    for i := range sine {
        sine[i] = complex(-1.0, 0.0)
    }

In fact, we can make a carrier wave at any phase angle and amplitude by using a bit of trig.

    // angle is in radians - here we have
    // 1.5 Pi (3 Tau) or 270 degrees.
    var angle = pi * 1.5
    
    // amplitude controls the transmitted
    // strength of the carrier wave.
    var amplitude = 1.0

    // output buffer as above
    var sine []complex{}
    
    for i := range sine {
        sine[i] = complex(
            amplitude*cos(angle),
            amplitude*sin(angle),
        )
    }

The amplitude of the transmitted wave is the absolute value of the IQ sample (sometimes called magnitude), and the phase can be computed as the angle (or argument). The amplitude remains constant (at 1) in both cases. Remember back to the airplane propeller or water droplets – we’re controlling where we’re observing the sine wave. It looks like a consistent value to us, but in reality it’s being transmitted as a pure carrier wave at the provided frequency. Changing the angle of the number we’re transmitting will control where in the sine wave cycle we’re “observing” it at.

Generating BPSK modulated IQ data

Modulating our carrier wave with our symbols is fairly straightforward to do – we can multiply the symbol by 1 to get the real value to be used in the IQ stream. Or, more simply - we can just use the symbol directly in the constructed IQ data.

  var sampleRate = 2,621,440
  var baudRate = 1024
  
  // This represents the number of IQ samples
  // required to send a single symbol at the
  // provided baud and sample rate. I picked
  // two numbers in order to avoid half samples.
  // We will transmit each symbol in blocks of
  // this size.
  var samplesPerSymbol = sampleRate / baudRate
  var samples = make([]complex, samplesPerSymbol)
  
  // symbol is one of 1, -1 or 0.
  for each symbol in symbols {
      for i := range samples {
          samples[i] = complex(symbol, 0)
      }
      // write the samples out to an output file
      // or radio.
      write(samples)
  }

If you want to check against a baseline capture, here’s 10 example packets at 204800 samples per second.

Next Steps

Now that we can transmit data, we’ll start working on a receive path in Part 3, in order to check our work when transmitting the packets, as well as being able to hear packets we transmit from afar, coming up next in Part 3!!