ma: add a bunch of graphs, some dsss demod proto explanation

ma/safety_reset.tex

@@ -1093,12 +1093,24 @@ interface and its good tolerance of system resets due to unexpected power loss.
 
 \begin{figure}
     \centering
-    \includegraphics{../lab-windows/fig_out/freq_meas_spectrum}
-    \caption{Fourier transform of the 24 hour grid frequency trace in fig. \ref{freq_meas_trace} with some notable peaks
-    annotated with the corresponding period in seconds. The $\frac{1}{f}$ line indicates a pink noise spectrum. We can
-    clearly see the noise spectrum flattens below some frequency around $\frac{1}{120 \text{s}}$. This effect is due to
-    primary control actively regulating grid frequency over such time intervals. Beyond the $\frac{1}{f}$ slope starting
-    at around $1 \text{Hz}$ we can make out a white noise floor in the order of $\frac{\mu\text{Hz}}{\text{Hz}}$.
+    \includegraphics{../lab-windows/fig_out/mains_voltage_spectrum}
+    \caption{Power spectral density of the mains voltage trace in fig. \ref{freq_meas_trace}. We can see the expected
+    peak at $50 \text{Hz}$ along with smaller peaks at odd harmonics. We can also see a number of spurious tones both
+    between harmonics and at low frequencies, as well as some bands containing high noise energy around $0.1
+    \text{Hz}$. This graph demonstrates a high signal-to-noise ratio that is not very demanding on our frequency
+    estimation algorithm.
+    }
+    \label{mains_voltage_spectrum}
+\end{figure}
+
+\begin{figure}
+    \centering
+    \includegraphics[width=\textwidth]{../lab-windows/fig_out/freq_meas_spectrum}
+    \caption{Power spectral density of the 24 hour grid frequency trace in fig. \ref{freq_meas_trace} with some notable
+    peaks annotated with the corresponding period in seconds. The $\frac{1}{f}$ line indicates a pink noise spectrum.
+    Around a period of $20 \text{s}$ the PSD starts to fall off at about $\frac{1}{f^3}$ until we can make out some
+    bumps at periods around $2$ and $3 \text{s}$.  Starting at at around $1 \text{Hz}$ we can see a white noise floor in
+    the order of $\frac{\mu\text{Hz}^2}{\text{Hz}}$.
     % TODO: where does this noise floor come from? Is it a fundamental property of the grid? Is it due to limitations of
     % our measurement setup (such as ocxo stability/phase noise) ???
     }
@@ -1120,9 +1132,130 @@ our target platform is a cheap low-end microcontroller. Our demonstrator firmwar
 language such as C or rust. For prototyping these languages lack flexibility compared to python.
 % FIXME introduce project outline, specs -> proto -> demo above!
 
-To validate our modulation scheme we performed a series of simulations. We produced modulated frequency data that we
-superimposed with an actual grid frequency measurement series.
-% FIXME do test series with simulated noise emulating measured noise spectrum
+To validate our modulation scheme we first performed a series of simulations on our python demodulator prototype
+implementation. To simulate a modulated grid frequency signal we added noise to a synthetic modulation signal. For most
+simulations we used measured frequency data gathered with our frequency sensor. We only have a limited amount of capture
+data. Re-using segements of this data as background noise in multiple simulation runs could hypothetically lead to our
+simulation results depending on individual features of this particular capture that would be common between all runs. To
+estimate the impact of this problem we re-ran some of our simulations with artificial random noise synthesized with a
+power spectral density matching that of our capture. To do this, we first measured our capture's PSD, then fitted a
+low-resolution spline to the PSD curve in log-log coordinates. We then generated white noise, multiplied the resampled
+spline with the DFT of the synthetic noise and performed an iDFT on the result. The resulting time-domain signal is our
+synthetic grid frequency data. Fig.\ \ref{freq_meas_spectrum} shows the PSD of our measured grid frequency signal. The
+red line indicates the low-resolution log-log spline interpolation used for shaping our artificial noise. Fig.\
+\ref{simulated_noise_spectrum} shows the PSD of our simulated signal overlayed with the same spline as a red line and
+shows time-domain traces of both simulated (blue) and reference signals (orange) at various time scales. Visually both
+signals look very similar, suggesting we have found a good synthetic approximation of our measurements.
+
+\begin{figure}
+    \centering
+    \includegraphics[width=\textwidth]{../lab-windows/fig_out/simulated_noise_spectrum}
+    \caption{Synthetic grid frequency in comparison with measured data. The topmost graph shows the synthetic spectrum
+    compared to the spline approximation of the measured spectrum (red line). The other graphs show time-domain
+    synthetic data (blue) in comparison with measured data (orange).
+    }
+    \label{simulated_noise_spectrum}
+\end{figure}
+
+In our simulations, we manipulated four main variables of our modulation scheme and demodulation algorithm and observed
+their impact on symbol error rate (SER):
+\begin{description}
+    \item[Modulation amplitude.] Higher amplitude should correspond to a lower SER.
+    \item[Modulation bit count.] Higher bit count $n$ means longer transmissions but yields higher theoretical decoding
+        gain, and should increase demodulator sensitivity. Ultimately, we want to find a sweet spot of manageable
+        transmission length at good demodulator sensitivity.
+    \item[Decimation] or DSSS chip duration. The chip time determines where in the grid frequency spectrum (fig.\
+        \ref{freq_meas_spectrum} our modulated signal is located. Given our noise spectrum (fig.\
+        \ref{freq_meas_spectrum}) lower chip durations (shifting our signal upwards in the spectrum) should yield lower
+        in-band background noise which should correspond to lower symbol error rates.
+    \item[Demodulation correlator peak threshold factor.] The first step of our prototype demodulation algorithm is to
+        calculate the correlation between all $2^n+1$ Gold sequences
+        % FIXME add a \ref here, describe proto demod alg somewhere
+        and to identify peaks corresponding to the input data containing a correctly aligned Gold sequence. The
+        threshold factor is a factor peaks of what magnitude compared to baseline noise levels are considered in the
+        following maximum likelihood estimation (MLE) decoding. % FIXME do we actually do MLS?
+\end{description}
+
+As indicated by our results, symbol error rate is a good proxy of demodulation performance. With decreasing
+signal-to-noise ratio, margins in various parts of the demodulator decrease which statistically leads to an increased
+symbol error rate. Our simulations yield smooth, reproducible SER curves with adequately low error bounds. This
+indicates SER is related fairly monotonically to the signal-to-noise margins inside our demodulator prototype.
+
+\begin{figure}
+    \centering
+    \includegraphics[width=\textwidth]{../lab-windows/fig_out/dsss_gold_nbits_overview}
+    \caption{
+    }
+    \label{dsss_gold_nbits_overview}
+\end{figure}
+
+\begin{figure}
+    \centering
+    \includegraphics[width=\textwidth]{../lab-windows/fig_out/dsss_gold_nbits_sensitivity}
+    \caption{
+    }
+    \label{dsss_gold_nbits_sensitivity}
+\end{figure}
+
+\begin{figure}
+    \begin{subfigure}{\textwidth}
+        \centering
+        \includegraphics[width=\textwidth]{../lab-windows/fig_out/dsss_thf_amplitude_5678}
+        \label{dsss_thf_amplitude_5678}
+        \caption{
+        }
+    \end{subfigure}
+    \begin{subfigure}{\textwidth}
+        \centering
+        \includegraphics[width=\textwidth]{../lab-windows/fig_out/dsss_thf_sensitivity_5678}
+        \label{dsss_thf_sensitivity_5678}
+        \caption{
+        }
+    \end{subfigure}
+    \caption{
+    }
+    \label{dsss_thf_sensitivity}
+\end{figure}
+
+\begin{figure}
+    \begin{subfigure}{\textwidth}
+        \centering
+        \includegraphics[width=\textwidth]{../lab-windows/fig_out/chip_duration_sensitivity_5}
+        \label{chip_duration_sensitivity_5}
+        \caption{
+        }
+    \end{subfigure}
+    \begin{subfigure}{\textwidth}
+        \centering
+        \includegraphics[width=\textwidth]{../lab-windows/fig_out/chip_duration_sensitivity_6}
+        \label{chip_duration_sensitivity_6}
+        \caption{
+        }
+    \end{subfigure}
+    \caption{
+    }
+    \label{chip_duration_sensitivity}
+\end{figure}
+
+\begin{figure}
+    \begin{subfigure}{\textwidth}
+        \centering
+        \includegraphics[width=\textwidth]{../lab-windows/fig_out/chip_duration_sensitivity_cmp_meas_6}
+        \label{chip_duration_sensitivity_cmp_meas_6}
+        \caption{
+        }
+    \end{subfigure}
+    \begin{subfigure}{\textwidth}
+        \centering
+        \includegraphics[width=\textwidth]{../lab-windows/fig_out/chip_duration_sensitivity_cmp_synth_6}
+        \label{chip_duration_sensitivity_cmp_synth_6}
+        \caption{
+        }
+    \end{subfigure}
+    \caption{
+    }
+    \label{chip_duration_sensitivity_cmp}
+\end{figure}
 
 \section{Implementation of a demonstrator unit}