GFSK Demodulation in Verilog on the DE1-SoC¶
V. Hunter Adams (vha3)¶
from IPython.display import Latex
import matplotlib.pyplot as plt
%matplotlib inline
import numpy
from numpy import genfromtxt
from mpl_toolkits.mplot3d import Axes3D
from numpy.linalg import pinv
from numpy.linalg import inv
from scipy.stats import norm
from scipy import integrate
from IPython.display import Image
from IPython.core.display import HTML
from IPython.display import HTML
Introduction¶
The objective of this project was to use the DE1-SoC to do realtime filtering, demodulation, decoding, and visualization of radio frequency transmissions from a CC1310 radio chip. A picture of the final setup is shown below.
Image(filename = "./setup.png", width=400, height=800)
High-Level Design¶
Rational and Sources for Idea¶
My graduate work surrounds the hardware and software design, launch, and operation of very small (5 cm x 5 cm) satellites (see below). Each of these chip-satellites is equipped with a 10 dBm transmitter that, with sufficient signal processing on the ground, can be heard from orbit. For previous missions, we have used a software-defined radio plugged into a laptop and performed all signal processing on the PC. This works well for a handful of chips but, ideally, I would like to be able to visualize data streaming from an entire swarm in realtime using a device that would fit into a backpack. This calls for lots of speed and lots of parallelization, making it a perfect FPGA project.
Image(filename = "./monarch.png", width=300, height=800)
Background Math¶
The RTL-SDR samples at 1 MHz, giving the receiver a bandwidth of 1 MHz (since each sample includes both in-phase and quadrature data). The CC1310 transmitters have a baud rate of 50 bits per second, giving me 20 samples per bit and a transmission bandwidth of 50 kHz. Thus, prior to demodulating the received signals, the FPGA must low-pass the baseband data down to 50 kHz. This is accomplished with a Finite Impulse Response filter.
Finite Impulse Response Filter¶
A finite impulse response filter is essentially just a weighted average of the $N$ most recent samples, where the weights associated with each of those samples are chosen in order to get a desired frequency response in the filter. For example, the output $y[n]$ of an FIR filter of order $N$ is shown below, where $b_i$ is the weight associated with sample $x_i$.
This filters have the advantage of requiring no feedback, being inherently stable, and being capable of being designed such that they are linear phase. This is accomplished by simply making the coefficient sequence symmetric, and is a desirable property for phase-sensitive applications like GFSK demodulation.
Many methods exist for coming up with a set of coefficients that meet requirements on passband ripple, stopband ripple, and transition width. Some of these include the window design method, frequency sampling method, and Parks-McClellan method. I used a Matlab tool that implements the Parks-McClellan method in order to come up with a set of coefficents for a 10-tap FIR filter that low-passes the raw RF samples down to 50 kHz. Such tools did not exist for any of the other aspects of this project.
GFSK Demodulation¶
The low-power transmitters (TI-CC1310's) use Gaussian Frequency Shift Keying (GFSK) to encode information at the carrier frequency. With GFSK, a logical 1 is encoded by increasing the frequency of the transmission to slightly greater than the carrier frequency and a logical zero is encoded by decreasing the frequency of the transmission to slightly less than the carrier frequency. This is in contrast to Amplitude Modulation which obviously modulates the amplitude in order to encode 1's and 0's, and Phase Modulation, which modulates the phase of the transmission (while keeping the frequency constant) in order to encode 1's and 0's. A good introductory article on these modulation schemes can be found here: https://www.allaboutcircuits.com/textbook/radio-frequency-analysis-design/radio-frequency-demodulation/quadrature-frequency-and-phase-demodulation/.
A discussion of the demodulation method requires a brief discussion of how the RTL-SDR sampling works. The RTL-SDR has two voltage-controlled oscillators that oscillate at precisely the carrier frequency of the transmitter (915 MHz). One of these oscillators is 90 degrees out of phase from the other. The RF transmissions received by the antenna are mixed with these local oscillators in order to get the baseband transmission. By mixing the received transmissions with both the in-phase oscillator and the out-of-phase oscillator, the RTL-SDR is able to represent the received transmission as the sum of two out-of-phase 915 MHz waves. One of these waves is in-phase (I) and the other is out-of-phase (or "quadrature", Q). This I/Q data is a nice way to represent the received transmissions because it is independent of the carrier frequency, and it includes phase information (which would be impossible to recover with just one local oscillator).
With the I/Q data, one has all of the information necessary to demodulate any of the modulation schemes mentioned above. For Amplitude Modulation, the relevant quantity would be the amplitude of the received transmission ($\sqrt{I^2 + Q^2}$). For phase modulation, the relevant quantity is the phase of the received signal relative to the local oscillators $\left(\text{atan2}\left(\frac{I}{Q}\right)\right)$. For Frequency Modulation, the information is encoded on the derivative of the phase. A procedural way to approximate this quantity is to find the conjugate product of the $n^{th}$ and $(n-1)^{st}$ samples (a complex number), and then to find the argument of the resulting complex number. If these two samples have the same phase, then the product will be a real number with argument 0. If these two samples are 90 degrees out of phase, then the product will be a purely imaginary number with argument $\frac{\pi}{2}$. The I/Q plot for a frequency-modulated signal ends up forming a circle, since the phase of the received transmission moves continually around the complex plane. For phase-modulated signals, the I/Q plots look like a collection of dots. Letting $\tilde{x}[n-1]$ be the complex conjugate of sample $x[n-1]$, this is represented by the below equation.
When no transmission is being received, the output of this demodulation method is white noise, since two consecutive samples may be any amount of phase separated from one another. During a transmission, however, this demodulation method is capable of recovering the logical waveform (the 1's and 0's) of the transmission. Below, the red trace is the output of the GFSK demodulation during a transmission. The logical 1's and 0's, clearly visible in the red trace, are represented by the blue binary-slicer below the red trace. In the upper-right corner, I have plotted the raw I and Q data (I on the horizontal, Q on the vertical). You can see that, during the transmission, this data forms a circle.
Image(filename = PATH + "iq.png", width=800, height=800)