modulation, freq estimation: add some blurb, citations

1 files changed, 48 insertions, 23 deletions

diff --git a/ma/safety_reset.tex b/ma/safety_reset.tex
index 3462740..303b519 100644
--- a/ma/safety_reset.tex
+++ b/ma/safety_reset.tex
@@ -1011,30 +1011,45 @@ controllable load:
     \item[Modulation amplitude] proportionally related to modulation power. In a practical setup we might realize a
         modulation power up to a few hundred \si{\mega\watt} which would yield maybe a few tens of \si{\milli\hertz} of
         frequency amplitude.
-    \item[Modulation pre-emphasis and slew-rate control]. Pre-emphasis might be necessary to ensure an adequate SNR at
-        the receiver. Slew-rate control and other shaping measures might be necessary to reduce the impact of these
-        sudden load changes on the transmitter's primary function (say, aluminium smelting) and to prevent disturbances
-        to grid components.
+    \item[Modulation pre-emphasis and slew-rate control]. Pre-emphasis might be necessary to ensure an adequate
+        Signal-to-Noise ratio (SNR) at the receiver. Slew-rate control and other shaping measures might be necessary to
+        reduce the impact of these sudden load changes on the transmitter's primary function (say, aluminium smelting)
+        and to prevent disturbances to grid components.
     \item[Modulation frequency]. For a practical implementation a careful study would be necessary to determine an
         optimal frequency band for operation. On one hand we need to prevent disturbances to the grid such as through
-        excitation of some local or inter-area modes. On the other hand we need to optimize SNR and data rate to achieve
-        optimal latency between transmission start and successful reception and to reduce the overall burden on
-        transmitter and grid.
+        excitation of some local or inter-area modes. On the other hand we need to optimize Signal-to-Noise ratio (SNR)
+        and data rate to achieve optimal latency between transmission start and successful reception and to reduce the
+        overall burden on transmitter and grid.
     \item[Further modulation parameters]. The modulation itself has numerous parameters that are discussed in sec.\
         \ref{mod_params} below.
 \end{description}
 
-\section{From grid frequency to a reliable communications channel}
-% FIXME
+\section{From grid frequency to a reliable communication channel}
 
 \subsection{Channel properties}
-% FIXME
+In this section we will explore how we can construct a reliable communication channel from the analog primitive we
+outline in the previous section. Our load control approach to grid frequency modulation leads to a channel with the
+following properties.
 
+\begin{description}
+    \item[Slow-changing.] Accurate grid frequency measurements need several periods of the mains sine wave. Faster
+        sampling rates can be achieved with more complex specialized synchrophasor estimation algorithms but this will
+        result in a tradeoff between sampling rate and accuracy\cite{belega01}.
+    \item[Analog.] Grid frequency is an analog signal.
+    \item[Noisy.] While stable over long periods of time thanks to Load-Frequency Control\cite{entsoe04} it shows
+        considerable random short-term variations. In addition our modulation amplitude is limited by technical and
+        economic constraints so we have to find a system that will work at poor SNRs.
+    \item[Polarized.] Grid frequency measurements have an inherent sense of \emph{up} (higher frequencies). We can use
+        this in a polarized modulation scheme to encode information without first transmitting some reference signal to
+        establish this polarization.
+\end{description}
 
 \subsection{Modulation and its parameters}
-
 \label{mod_params}
 
+In this section we will consider how to select a good set of parameters for a modulation scheme fitting grid frequency
+modulation.
+
 The sensitivity of the grid to oscillation at particular frequencies described above means we should avoid any
 modulation technique that would concentrate a lot of energy in a small bandwidth. Taking this principle to its extreme
 provides us with a useful pointer towards techniques that might work well: Spread-spectrum techniques. By employing
@@ -1285,12 +1300,12 @@ required precision for manageable averaging times--we would need either a ADC sa
 for a reconstruction through interpolated readings an impractically high ADC resolution.
 
 Detail on the inner workings of commercial phasor measurement units is scarce but given their essential role to SCADA
-systems there is a large amount of academic research on such algorithms\cite{narduzzi01,derviskadic01}. A popular
-approach to these systems is to perform a Short-Time Fourier Transform (STFT) on ADC data sampled at high sampling rate
-(e.g. \SI{10}{\kilo\hertz}) and then perform some analysis on the frequency-domain data to precisely locate the strong peak
-around \SI{50}{\hertz}. A key observation here is that FFT bin size is going to be much larger than required frequency
-resolution. This fundamental limitiation follows from the nyquist criterion %FIXME maybe cite?  and if we had to process
-an \emph{arbitrary} signal this would highly limit our practical measurement accuracy
+systems there is a large amount of academic research on such algorithms\cite{narduzzi01,derviskadic01,belega01}. A
+popular approach to these systems is to perform a Short-Time Fourier Transform (STFT) on ADC data sampled at high
+sampling rate (e.g. \SI{10}{\kilo\hertz}) and then perform some analysis on the frequency-domain data to precisely
+locate the strong peak around \SI{50}{\hertz}. A key observation here is that FFT bin size is going to be much larger
+than required frequency resolution. This fundamental limitiation follows from the nyquist criterion %FIXME maybe cite?
+and if we had to process an \emph{arbitrary} signal this would highly limit our practical measurement accuracy
 \footnote{
     Some software packages providing FFT or STFT primitives such as scipy\cite{virtanen01} allow the user to
     super-sample FFT output by specifying an FFT width larger than input data length, padding the input data with zeros
@@ -1351,12 +1366,22 @@ resolution and despite numerous distortions.
 Published grid frequency estimation algorithms such as \textcite{narduzzi01} or \textcite{derviskadic01} are rather
 sophisticated and use a combination of techniques to reduce numerical errors in FFT calculation and peak fitting. Given
 that we do not need reference standard-grade accuracy for our application we chose to start with a very basic algorithm
-instead. We chose to use a general approach developed by experimental physicists at CERN that is described by
-\textcite{gasior01}. This approach assumes a general sinusoidal signal superimposed with harmonics and broadband noise.
-Applicable to a wide spectrum of practical signal analysis tasks it is a reasonable first-degree approximation of the
-much more sophisticated estimation algorithms developed specifically for power systems.  Some algorithms have components
-such as kalman filters\cite{narduzzi01} that require a phyiscal model.  As a general algorithm from \textcite{gasior01}
-does not require this kind of application-specific tuning, eliminating one source of error.
+instead. We chose to use a general approach to estimate the precise fundamental frequency of an arbitrary signal that
+was developed by experimental physicists at CERN and that is described by \textcite{gasior01}. This approach assumes a
+general sinusoidal signal superimposed with harmonics and broadband noise.  Applicable to a wide spectrum of practical
+signal analysis tasks it is a reasonable first-degree approximation of the much more sophisticated estimation algorithms
+developed specifically for power systems.  Some algorithms have components such as kalman filters\cite{narduzzi01} that
+require a phyiscal model.  As a general algorithm from \textcite{gasior01} does not require this kind of
+application-specific tuning, eliminating one source of error.
+
+The \textcite{gasior01} algorithm passes the windowed input signal through a DFT, then interpolates the signal's
+fundamental frequency by fitting a wavelet such as a gaussian to the largest peak in the DFT results. The bias parameter
+of this curve fit is an accurate estimation of the signal's fundamental frequency. This algorithm is similar to the
+simpler interpolated DFT algorithm used as a reference in much of the synchrophasor estimation
+literature\cite{borkowski01}. The three-term variant of the maximum sidelobe decay window often used there is a blackman
+window with parameter $\alpha = \frac{1}{4}$. Analysis has shown\cite{belega01} that the interpolated DFT algorithm is
+worse than algorithms involving more complex models under some conditions but that there is \emph{no free lunch} meaning
+that more complex perform worse when the input signal deviates from their models.
 
 \subsection{Frequency sensor hardware design}
 \label{sec-fsensor}