Merge remote-tracking branch 'origin/master'

1 files changed, 4 insertions, 4 deletions

diff --git a/doc/thesis/chapters/implementation.tex b/doc/thesis/chapters/implementation.tex
index c22f6d3..0863cf9 100644
--- a/doc/thesis/chapters/implementation.tex
+++ b/doc/thesis/chapters/implementation.tex
@@ -90,7 +90,7 @@ As receivers and transmitter devices for the SDR setup two USRP B210 devices fro
 	}
 \end{figure}
 
-To compute the empirical bit error rate (BER) of the setup, the data has to be framed by the transmitter and the bitstream synchronized on the receiver side. The structure of a data packet used in the implementation is shown in \figref{fig:dataframe}. A frame begins with an user specified \(k\)-byte preamble, that in the current implementation serves as synchronization pattern. Another use case for the preamble sequence could be to introduce channel estimation pilot symbols. Following the preamble are 4 bytes encoded using a (31, 26) Hamming code (plus 1 padding bit), that contain metadata about the packet, namely payload ID and payload length. Because the payload length in bytes is encoded in 21 bits, the theoretical maximum payload size is 2 MiB, which together with 32 possible unique IDs gives a maximum data transfer with unique frame headers of 64 MiB. These constraints are a result of decisions made to keep the implementation simple.
+To compute the empirical bit error rate (BER) of the setup, the data has to be framed by the transmitter and the bitstream needs to be synchronized on the receiver side. The structure of a data packet used in the implementation is shown in \figref{fig:dataframe}. A frame begins with an user specified \(k\)-byte preamble, that in the current implementation serves as synchronization pattern. Another use case for the preamble sequence could be to introduce channel estimation pilot symbols. Following the preamble are 4 bytes encoded using a (31, 26) Hamming code (plus 1 padding bit), that contain metadata about the packet, namely payload ID and payload length. Because the payload length in bytes is encoded in 21 bits, the theoretical maximum payload size is 2 MiB, which together with 32 possible unique IDs gives a maximum data transfer with unique frame headers of 64 MiB. These constraints are a result of decisions made to keep the implementation simple. % TODO: explain why its simpler this way
 
 
 \subsection{Modulation}
@@ -113,9 +113,9 @@ What is here referred to as envelope detector has the purpose of synchronizing t
 
 \subsection{Frame synchronization and phase correction} \label{sec:phasecorr}
 
-Once the envelope's clock is synchronized in the processing chain the data stream has one sample per symbol. At this point it is necessary to find where each data frame starts or end in order to correctly decode their payloads. For such purpose a special sequence called \emph{access code} is put in front of each frame. Access codes are sequences of samples that are carefully constructed to have an autocorrelation with a high peak at zero, and that rapidly decreases for increasing shifts. In other words, the autocorrelation of an access code high only when the sequence is perfectly aligned. Thus by cross correlating an envelope signal \(r(t)\), that periodically contains an access code \(a(t)\) with the access code itself, and looking for peaks in the result, it is possible to determine where each frame begin. Furthermore by analyzing the values of the peaks it is possible to extract information about the phase and frequency offsets.
+Once the envelope's clock is synchronized in the processing chain the data stream has one sample per symbol. At this point it is necessary to find where each data frame starts or end in order to correctly decode their payloads. For such purpose a special sequence called \emph{access code} is put in front of each frame. Access codes are sequences of samples that are carefully constructed to have an autocorrelation with a high peak at zero, and that rapidly decreases for increasing shifts. In other words, the autocorrelation of an access code high only when the sequence is perfectly aligned. Thus, by cross correlating an envelope signal \(r(t)\), that periodically contains an access code \(a(t)\) with the access code itself, and looking for peaks in the result, it is possible to determine where each frame begin. Furthermore, by analyzing the values of the peaks it is possible to extract information about the phase and frequency offsets.
 
-To understand how correlation peaks allow for fine phase correction, recall that mathematically the cross correlation (denoted here by \(\star\)) of two complex valued signals is
+To understand how correlation peaks allow for fine phase correction, recall that the cross correlation (denoted here by \(\star\)) of two complex valued signals is
 \begin{equation}
 	R_{ra}
 	= (r \star a)(t)
@@ -149,7 +149,7 @@ To implement in GR what was discussed in section \ref{sec:phasecorr} two blocks
 
 Tags are GR's way of working with metadata that is attached to a sample. Internally tags are just polymorphic data structures containing a number indicating the absolute offset (in samples), and a pair of arbitrary values called ``key'' and ``value''. Tags are passed on from one block to the next like sample streams (unless the block specifies to do otherwise).
 
-Thus the tagged stream is processed with a custom block, of which a simplified version of its work function shown in listing \ref{lst:phasecorr-work}. The custom block also implements fine frequency correction (shown in listing \ref{lst:phasecorr-blockphase}) by linearly interpolating the phase estimates between each pair of tags (called chunk). Mathematically this can be rather trivially be formulated for a chunk of \(N\) samples with the
+Thus, the tagged stream is processed with a custom block, of which a simplified version of its work function shown in listing \ref{lst:phasecorr-work}. The custom block also implements fine frequency correction (shown in listing \ref{lst:phasecorr-blockphase}) by linearly interpolating the phase estimates between each pair of tags (called chunk). This can be rather trivially be formulated for a chunk of \(N\) samples with the
 \begin{subequations}
 	\begin{align}
 		k\text{-th chunk digital frequency} \quad  & \Omega_k = (\varphi_{k+1} - \varphi_k) / N, \text{ and the }\\