More corrections

5 files changed, 53 insertions, 55 deletions

diff --git a/doc/thesis/Fading.tex b/doc/thesis/Fading.tex
index 305ffde..82ee86d 100644
--- a/doc/thesis/Fading.tex
+++ b/doc/thesis/Fading.tex
@@ -104,7 +104,7 @@
 	\include{tex/titlepage}
 
 	\begin{abstract}
-		Today wireless devices are ubiquitous and have to work under technically adverse circumstances. Among the multitude of problems affecting wireless communication systems, a common issue is what is known as \emph{multipath fading effect}. The aim of this work is to create a demonstrative platform that can illustrate how multipath fading degrades a wireless communication link. A number of scenarios affected by multipath fading were elaborated, simulated and measured using two USRP B210 software defined radios and the GNU Radio signal processing toolkit. To demonstrate the effects, a custom graphical front-end for GNU Radio's signal processing was built using the Dear IMGUI framework.
+		Today wireless devices are omnipresent and have to work under technically adverse circumstances. Among the multitude of problems affecting wireless communication systems, a common issue is what is known as the \emph{multipath fading effect}. The aim of this work is to create a demonstrative platform that can illustrate how multipath fading degrades a wireless communication link. A number of scenarios affected by multipath fading were elaborated, simulated and measured using two USRP B210 software defined radios and the GNU Radio signal processing toolkit. To demonstrate the effects, a custom graphical front-end for GNU Radio was built using the Dear IMGUI framework.
 	\end{abstract}
 
 	\cleardoublepage
@@ -125,9 +125,8 @@
 		j                  & Imaginary unit \\
 		\phi^*             & Complex conjugate of \(\phi\) \\
 		\vec{v},\, \vec{m} & Vector quantity \\
-		\E{X}              & Expectation of a random variable \(X\) \\
 		\midrule
-		\multicolumn{2}{l}{\itshape Physical quantities} \\
+		\multicolumn{2}{l}{\itshape Physical quantities or constants} \\
 		t       & Time \\
 		T       & Period \\
 		\tau    & Convolution time \\
@@ -171,7 +170,7 @@
 		API    & Application Programming Interface                \\
 		AWGN   & Additive White Gaussian Noise                    \\
 		BER    & Bit Error Rate                                   \\
-		CAZAC  & Constant Amplitude Zero Autocorrelation Waveform \\
+		CAZAC  & Constant Amplitude Zero Autocorrelation waveform \\
 		CIR    & Channel Impulse Response                         \\
 		CMA    & Constant Modulus Adaptive filter                 \\
 		DPG    & Dear PyGUI                                       \\
@@ -193,8 +192,8 @@
 		PSAM   & Pilot Symbols Assisted Modulation                \\
 		PSK    & Phase Shift Keying                               \\
 		QAM    & Quadrature Amplitude Modulation                  \\
-		QPSK   & Quadrature Phase Shift Keying                    \\
-		SDR    & Software Defined radio                           \\
+		QPSK   & Quaternary Phase Shift Keying                    \\
+		SDR    & Software Defined Radio                           \\
 		UDP    & User Datagram Protocol                           \\
 		US     & Uncorrelated Scattering                          \\
 		WSS    & Wide Sense Stationary                            \\
diff --git a/doc/thesis/chapters/conclusions.tex b/doc/thesis/chapters/conclusions.tex
index 2852639..80776f9 100644
--- a/doc/thesis/chapters/conclusions.tex
+++ b/doc/thesis/chapters/conclusions.tex
@@ -22,7 +22,7 @@ In addition to fixing the issue discussed in section \ref{sec:gui-issue-single-t
 %TODO : 
 
 
-\subsection{Channel parameters estimation with PSAM}
+\subsection{Channel parameters estimation with PSAM} \label{sec:psam}
 %TODO: Picture send mention
 
 An interesting continuation of this work could be to regularly interpolate some so called pilot symbols in the modulated data stream. In short, the pilot symbol assisted modulation (PSAM) technique consists of periodically inserting informationless (known) symbols in the data stream, which can then be used to estimate the fading parameters of the communication channel. More details are presented in \cite{Xiaoyi1999} (and its references) from which the illustrations in \figref{fig:psam} were taken.
diff --git a/doc/thesis/chapters/implementation.tex b/doc/thesis/chapters/implementation.tex
index 50485b9..1e68a49 100644
--- a/doc/thesis/chapters/implementation.tex
+++ b/doc/thesis/chapters/implementation.tex
@@ -4,15 +4,15 @@
 
 \section{Overview}
 
-To illustrate the effects of multipath fading on a wireless transmission link, two machines with an USRP B210 each (hardware) were set up to run a software defined radio (SDR) using the GNU Radio (GR) signal processing toolkit. The latter can also be used as standalone (without hardware) to simulate a channel model. Therefore, simulations of different scenarios affected by multipath fading were elaborated and constructed using a static FIR model\footnote{As discussed in sections \ref{sec:discrete-time-model} and \ref{sec:fractional-delay}} and dynamic statistical model\footnote{See section \ref{sec:statistical-model}}. To present the results a graphical interface was made using the Dear IMGUI framework.
+To illustrate the effects of multipath fading on a wireless transmission link, two machines with an USRP B210 each (hardware) were set up to run a software defined radio (SDR) using the GNU Radio (GR) signal processing toolkit. The latter can also be used as standalone (without hardware) to simulate a channel model. Therefore, simulations of different scenarios affected by multipath fading were elaborated and constructed first using a static FIR model\footnote{As discussed in sections \ref{sec:discrete-time-model} and \ref{sec:fractional-delay}}, and then with a dynamic statistical model\footnote{See section \ref{sec:statistical-model}}. To present the results a graphical interface was made using the Dear IMGUI framework.
 
-The rest of the chapter is structured as follows. First the tools used in this project are introduced. Then implementations of the transmitter and receiver chains is explained. Subsequently simulations and measurements of fading channels with an empirically computed bit error rate (BER) are presented. Finally, the issues and the state of our current implementation is discussed.
+The rest of the chapter is structured as follows. First the tools used in this project are introduced. Then implementations of the transmitter and receiver chains is explained. Subsequently simulations and measurements of fading channels with their respective empirically computed bit error rate (BER) are presented. Finally, the current state of the implementation and issues are discussed.
 
 \section{Software Stack}
 
 \subsection{GNU Radio}
 
-For both the signal processing and the simulations the GNU Radio (GR) toolkit was chosen, as it already had drivers for the USRP hardware. GR is an open-source free software framework that can be used to build signal processing chains and SDRs. GR is composed of two parts: a C\texttt{++} library with Python bindings to write signal processing code, and GNU Radio Companion (GRC). GRC is a graphical user interface made to more easily construct signal processing chains. Signal processing algorithms as ``blocks'' that are chained together with arrows, essentially drawing a diagram called ``flow graph''. An example of a flow graph can be seen in \figref{fig:sync-lock-flowgraph}.
+For both the signal processing and the simulations the GNU Radio (GR) toolkit was chosen, as it already had integrations with the USRP hardware drivers. GR is an open-source free software framework that can be used to build signal processing chains and SDRs. GR is composed of two parts: a C\texttt{++} library with Python bindings to write signal processing code, and GNU Radio Companion (GRC). GRC is a graphical user interface made to more easily construct signal processing chains. Signal processing algorithms are graphically shown as ``blocks'' that can be chained together with arrows, essentially drawing a diagram called ``flow graph''. An example of a flow graph can be seen in \figref{fig:sync-lock-flowgraph}.
 
 Internally GR works by keeping multiple memory buffers of samples, that are passed as pointers to the signal processing algorithms' ``work functions''. When the signal processing is complete, the output buffer of one block is given to the next block as input according to how they were connected in the flow graph. The structure of a block is shown in the Python listing \ref{lst:gr-block-py}. To improve performance GR creates a thread for each work function to parallelize the workload of the concurrently running signal processing blocks. For more details see the GNU Radio Wiki and User Manual in \cite{GRWiki}.
 
@@ -44,13 +44,11 @@ class myblock(gr.sync_block):
 
 To construct a graphical interface for a demonstration platform the Dear IMGUI (immediate mode graphical user interface) library was chosen, mainly for its ease of use, wide range of technical capabilites and high refresh rate. Dear PyGUI (DPG) are the Python bindings for the Dear IMGUI library.
 
-The DPG GUI communicates with the GR flow graphs using the IP/UDP protocol. This decision to separate the project into two parts that communicate over the IP network was made because it is not easy to extend the graphical interface of GRC without interfering with the sophisticated multi-threaded architecture of GR. Furthermore, this allows to have multiple correctly configured flow graphs on disk and to choose which one to run and display on the graphical interface, instead of having a single flow graph whose parameters need to be changed each time. As a side effect, in theory this setup allows to have one computer running the graphical interface, and another remote machine running just the flow graph.
+The DPG front-end communicates with the GR flow graphs using the IP/UDP protocol. This decision to separate the project into two parts that communicate over the IP network was made because it is not easy to extend the graphical interface of GRC without interfering with the sophisticated multi-threaded architecture of GR. Furthermore, this allows to have multiple correctly configured flow graphs on disk and to choose which one to run and display on the graphical interface, instead of having a single flow graph whose parameters need to be changed each time. As a side effect, theoretically this setup allows to have one computer running the graphical interface, and another remote machine running just the flow graph. Though the latency caused by the UDP/IP could be substantial.
 
 %TODO: Describe GUI Plott.
 \section{Hardware}
 
-As receivers and transmitter devices for the SDR setup two USRP B210 devices from Ettus Research were used. Some technical specifications are shown in \tabref{tab:usrp-specs}. GR provides off the shelf blocks that interface with the official API provided from Ettus Research.
-
 \begin{table}[h]
 	\centering
 	\begin{tabular}{ll}
@@ -58,13 +56,15 @@ As receivers and transmitter devices for the SDR setup two USRP B210 devices fro
 		Dimensions               & \(9.7 \times 15.5 \times 1.5\)\,cm \\
 		Ports                    & 2\,TX, 2\,RX, Half or Full Duplex   \\
 		RF frequencies           & \SI{70}{\mega\hertz} to \SI{6}{\giga\hertz} \\
-		Bandwidth                & \SI{200}{\kilo\hertz}--\SI{56}{\mega\hertz} \\
+		Bandwidth                & \SI{200}{\kilo\hertz} --- \SI{56}{\mega\hertz} \\
 		External reference input & \SI{10}{\mega\hertz}                         \\
 		\bottomrule
 	\end{tabular}
 	\caption{USRP B210 specifications \cite{EttusUSRPB210}. \label{tab:usrp-specs}}
 \end{table}
 
+As already mentioned, for the receiver and transmitter in the SDR setup two USRP B210 devices from Ettus Research were used. Some technical specifications of the device are shown in \tabref{tab:usrp-specs}. GR provides off the shelf blocks that interface with the official API provided from Ettus Research.
+
 \section{Transmitter chain}
 
 \subsection{Data frame} \label{sec:data-frame}
@@ -78,7 +78,7 @@ 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 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 a 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
+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 a 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 (this will be discussed more in section \ref{sec:psam}). 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}
@@ -120,14 +120,14 @@ which is equivalent to a convolution, with the left term being time-reversed com
 	= \int_\mathbb{R} a(\tau) a^*(\tau - 0) \,d\tau
 	= \int_\mathbb{R} |a(\tau)|^2 \,d\tau \in \mathbb{R},
 \end{equation}
-which is a real number. And more importantly the correlation with an out of phase copy \(a'(t) = a(t) e^{j\varphi}\) at 0 is
+which is a real number. And more importantly the correlation with an out of phase copy \(a'(t) = a(t) e^{j\varphi}\) at \(t = 0\) is
 \begin{equation} \label{eqn:xc-oop-copy}
 	% R_{a'a} =
 	(a' \star a)(0) 
 	= \int_\mathbb{R} a(\tau)e^{j\varphi}  a^*(\tau) \,d\tau
 	= R_{aa} e^{j\varphi}.
 \end{equation}
-The relevant observation to make in \eqref{eqn:xc-oop-copy} that since \(R_{aa}\) is a real number, the phase of the cross correlation at \(t = 0\) is the phase of \(a'(t)\). This fact can be exploited to implement fine phase correction for the received envelope in relatively few steps as follows:
+The relevant observation to make in \eqref{eqn:xc-oop-copy} is that since \(R_{aa}\) is a real number, the phase of the cross correlation at \(t = 0\) is the phase of \(a'(t)\). This fact can be exploited to implement fine phase correction for the received envelope in relatively few steps as follows:
 \begin{enumerate}
 	\item Compute the cross correlation \(R_{ra}\) of the envelope \(r(t)\) with the access code \(a(t)\),
 	\item Find the maximum value of \(\hat{R}_{ra} = \max R_{ra}(t)\) (correlation peak),
@@ -139,18 +139,18 @@ The relevant observation to make in \eqref{eqn:xc-oop-copy} that since \(R_{aa}\
 	\centering
 	\includegraphics[width = .95\linewidth]{figures/screenshots/sync_lock}
 	\caption{
-		Part of the GNU Radio flow graph for the QPSK modulated link (with hardware). The shown blocks are used to synchronize, equalize and lock the envelope.
+		Part of the GNU Radio flow graph for the QPSK modulated link (with hardware). The shown blocks are used to equalize and lock the envelope.
 		\label{fig:sync-lock-flowgraph}
 	}
 \end{figure}
 
 \subsubsection{Implementing fine phase and frequency correction} \label{sec:implement-phasecorr}
 
-To implement in GR what was discussed in the previous section the two blocks shown in \figref{fig:sync-lock-flowgraph} were used: a correlator estimator block, and a custom block. The former essentially implements the first 3 of the steps discussed at the end of section \ref{sec:phasecorr}. The correlator estimator block is given a sequence of samples, and when the cross correlation between them and the input stream is higher than a certain threshold (90\% of the amplitude of a perfect autocorrelation), it produces a ``tag'' in the output stream, that contains the phase estimate.
+To implement in GR what was discussed above the two blocks shown in \figref{fig:sync-lock-flowgraph} were used: a correlator estimator block, and a custom block. The former essentially implements the first 3 of the steps discussed at the end of the previous subsection. The correlator estimator block is given a sequence of samples, and when the cross correlation between them and the input stream is higher than a certain threshold (90\% of the amplitude of a perfect autocorrelation), it produces a ``tag'' in the output stream, that contains the phase estimate.
 
 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). 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 as the
 \begin{subequations}
 	\begin{align}
 		k\text{-th chunk digital frequency} \quad  & \Omega_k = (\varphi_{k+1} - \varphi_k) / N, \text{ and the }\\
@@ -173,7 +173,7 @@ Because the frequency estimate is linearly interpolated, the phase error may not
 	\implies T = 1/f_s \leq \frac{1}{2\Delta f N' \kappa},
 	\iff N' \leq \frac{1}{2\Delta f T \kappa},
 \end{equation}
-must hold. By further setting \(\kappa = 4\) and \(N' = 32\) we obtain a minimum sampling frequency of approximately \(\SI{618.5}{\kilo\hertz}\), or conversely by letting \(f_s = \SI{1}{\mega\hertz}\) we have a maximum frame length of \(N' = 51\) symbols. In other words, roughly every 50 symbols the system must send an access code sequence. This result is rather unfortunate as it requires a lot more processing power than expected.
+must hold. By further setting \(\kappa = 4\) and \(N' = 32\) we obtain a minimum sampling frequency of approximately \(\SI{618.5}{\kilo\hertz}\), or conversely by letting \(f_s = \SI{1}{\mega\hertz}\) we have a maximum frame length of \(N' = 51\) symbols. In other words, roughly every 50 symbols the system must send an access code sequence. This result is rather unfortunate as it means that more processing power than expected is required.
 
 \begin{figure}
 	\centering
diff --git a/doc/thesis/chapters/introduction.tex b/doc/thesis/chapters/introduction.tex
index 3e9e6d9..0429ea5 100644
--- a/doc/thesis/chapters/introduction.tex
+++ b/doc/thesis/chapters/introduction.tex
@@ -6,9 +6,9 @@
 
 It is undeniable that in the last two decades wireless devices have become extremely ubiquitous, and are no longer employed under carefully chosen conditions.
 
-Nowadays smart phones, internet of things (IoT) devices and many other wireless applications are omnipresent and have to work in environments that are very far from ideal. Furthermore, in addition to the already large class of networked appliances, next generation wireless devices in urban environments will include the new category of vehicles \cite{AntonescuTB17}, where reliability of inter-vehicular communication directly translates into safety. While at the same time in rural regions, developing countries as well as in other low-user density areas, wireless transmission links using mesh networks have become a practical alternative to wired broadband \cite{Macmillan2019tidal,Subramanian2006rethinking,Flickenger2007wireless}.
+Nowadays smart phones, internet of things (IoT) and many other wireless applications are omnipresent and have to work in environments that are very far from ideal. Furthermore, in addition to the already large class of networked appliances, the next generation of wireless devices in urban environments will include the new category of vehicles \cite{AntonescuTB17}. While at the same time in rural regions, developing countries as well as in other low-user density areas, wireless transmission links using mesh networks have become a practical alternative to wired broadband \cite{Macmillan2019tidal,Subramanian2006rethinking,Flickenger2007wireless}. Thus today the study of problems concerning wireless devices is a very relevant topic.
 
-Thus today the study of problems concerning wireless devices is a very relevant topic. In particular, a common issue observed in the previously mentioned use cases is the so called \emph{multipath fading effect}, that degrades the reliability of a wireless transmission link \cite{Mathis, Gallager}. The problem of fading was actually foreseen \cite{Frederiksen2002overview,Maddocks1993introduction} and today most modern transmission schemes implement measures to reduce the effects fading \cite{Mathis,Hsu}.
+In particular, a common issue observed in the previously mentioned use cases is the so called \emph{multipath fading effect}, that degrades the reliability of a wireless transmission link \cite{Mathis, Gallager}. The problem of fading was actually foreseen \cite{Frederiksen2002overview,Maddocks1993introduction} and today most modern transmission schemes implement measures to reduce the effects fading \cite{Mathis,Hsu}.
 
 This work studies the multipath fading effect, and how it affects modern digital transmission systems that use quadrature amplitude (QAM) and phase shift keying (PSK) modulation.
 
@@ -22,4 +22,4 @@ The entire task description is found in the appendix.
 
 \section{Overview}
 
-In chapter \ref{chp:theory}, the theoretical formulation and mathematical basis for the modulation schemes and channel models that have been used in this project are presented. Specifically \(M\)-ary QAM, \(M\)-PSK modulation, and three models for multipath fading are explained (continuous time, discrete time and statistical). Chapter \ref{chp:implementation} describes in detail our implementation. The transmitter and receiver chains are explained and simulations as well as measurements under different multipath fading conditions are presented. Finally some problems of the current implementation project are addressed. Chapter \ref{chp:conclusions} discusses the results of the project, and suggests how the device could be improved in the future.
+In chapter \ref{chp:theory}, the theoretical formulations and mathematical basis necessary to understand the problem analyzed by this work are presented. Specifically \(M\)-ary QAM, \(M\)-PSK modulation, and three models for multipath fading (continuous time, discrete time and statistical) are explained. Chapter \ref{chp:implementation} describes in detail our implementation. The transmitter and receiver chains are explained, and simulations as well as measurements under different multipath fading conditions are presented. Finally some problems of the current implementation are addressed. Chapter \ref{chp:conclusions} discusses the results of the project, and suggests how the device could be improved in the future.
diff --git a/doc/thesis/chapters/theory.tex b/doc/thesis/chapters/theory.tex
index c7547cd..87c5867 100644
--- a/doc/thesis/chapters/theory.tex
+++ b/doc/thesis/chapters/theory.tex
@@ -9,22 +9,22 @@
 	}
 	\caption{
 		Block diagram of the developed wireless communication system with annotated signal names. Frequency domain representations of signals use the uppercase symbol of their respective time domain name. Amplification constants in the channel (for \(s(t)\) and \(r(t)\)) were omitted throughout the document for readability.
-		\label{fig:notation}
+		\label{fig:blockdiagram}
 	}
 \end{figure}
 
 \section{Overview}
 
-The following two sections will briefly introduce mathematical formulations of the modulation schemes and of the channel models used in this project. The notation used is summarized in \figref{fig:notation}. For conciseness, encoding schemes and (digital) signal processing calculations are left out and discussed later.  Section \ref{sec:multipath-fading} presents an established mathematical model to understand multipath fading, as well as a brief description of a discrete-time model and the intricacies caused by the sampling process. Finally, the concept of stochastic models is mentioned, as they are often used to simulate multipath channels \cite{Messier,Mathis}.
+The sections \ref{sec:mqam} and \ref{sec:mpsk} will briefly introduce the mathematical formulations of the QAM and PSK modulation schemes. Section \ref{sec:multipath-fading} the channel models used in this project are explained. The setup is summarized as a block diagram in \figref{fig:blockdiagram}. For conciseness, encoding schemes and (digital) signal processing calculations are left out and discussed later. Section \ref{sec:multipath-fading} presents an established mathematical model to understand multipath fading, as well as a brief description of a discrete-time model and the intricacies caused by the sampling process. Finally, the concept of stochastic models is mentioned, as they are often used to simulate multipath channels \cite{Messier,Mathis}.
 
 %% TODO: A section on maths?
 % \section{Signal space and linear operators}
 
-\section{Quadrature amplitude modulation (\(M\)-ary QAM)}
+\section{Quadrature amplitude modulation (\(M\)-ary QAM)} \label{sec:mqam}
 
 \begin{figure}
 	\centering
-	\resizebox{\linewidth}{!}{
+	\resizebox{.9\linewidth}{!}{
 		\input{figures/tikz/qam-modulator}
 	}
 	\caption{
@@ -39,7 +39,7 @@ Quadrature amplitude modulation is a family of modern digital modulation methods
 
 \paragraph{Bit splitter}
 
-As mentioned above, quadrature modulation allows to send more than one bit per unit time. The first step is to use a so called bit splitter, that converts the continuous bit stream \(m(n)\) into pairs of chunks of \(\kappa = \log_2 \sqrt{M}\) bits each, where \(M\) is a power of 2. The two bit vectors of length \(\kappa\), denoted by \(\vec{m}_i\) and \(\vec{m}_q\) in figure \ref{fig:quadrature-modulation}, are called in-phase and quadrature component respectively\cite{Hsu}. The reason will become more clear later.
+As mentioned above, quadrature modulation allows to send more than one bit per unit time. The first step of the process, is to use a so called bit splitter, that converts the continuous bit stream \(m(n)\) into pairs of chunks of \(\kappa = \log_2 \sqrt{M}\) bits each, where \(M\) is a power of 2. The two bit vectors of length \(\kappa\), denoted by \(\vec{m}_i\) and \(\vec{m}_q\) in figure \ref{fig:quadrature-modulation}, are called in-phase and quadrature component respectively\cite{Hsu}. The reason will become more clear later.
 
 \paragraph{Binary to level converter}
 
@@ -54,12 +54,12 @@ i.e. a pulse function\footnote{Typically a root raised cosine to optimize for ba
 
 \paragraph{Mixer}
 
-Having analog level signals, it is now possible to mix them with radio frequency carriers. Because there are two waveforms, one might expect that two carrier frequencies are necessary, however this is not the case. The two component \(m_i(t)\) and \(m_q(t)\) are mixed with two different periodic signals \(\phi_i(t)\) and \(\phi_q(t)\) that have the same frequency \(\omega_c = 2\pi / T\). How this is possible is explained in the next section.
+By having analog level signals, it is now possible to mix them with radio frequency carriers. Because there are two waveforms, one might expect that two carrier frequencies are necessary, however this is not the case. The two component \(m_i(t)\) and \(m_q(t)\) are mixed with two different periodic signals \(\phi_i(t)\) and \(\phi_q(t)\) that have the same frequency \(\omega_c = 2\pi / T\). How this is possible is explained in the next section.
 
 
 \subsection{Orthogonality of carrier signals}
 
-Before explaining how the two carrier signals are generated, some important mathematical properties of \(\phi_i\) and \(\phi_q\) have to be discussed, in order to modulate two messages over the same frequency \(\omega_c\). The two carriers need to be \emph{orthonormal}\footnote{Actually orthogonality alone would be sufficient, however then the left side of \eqref{eqn:orthonormal-condition} would not equal 1, and an inconvenient factor would be introduced in many later equations \cite{Gallager,Hsu}.} to each other. Mathematically this is expressed by the conditions \cite{Gallager}
+Before explaining how the two carrier signals are generated, some important mathematical properties of \(\phi_i\) and \(\phi_q\) have to be discussed. In order to modulate two messages over the same frequency \(\omega_c\), the two carriers need to be \emph{orthonormal}\footnote{Actually orthogonality alone would be sufficient, however then the left side of \eqref{eqn:orthonormal-condition} would not equal 1, and an inconvenient factor would be introduced in many later equations \cite{Gallager,Hsu}.} to each other. Mathematically this is expressed by the conditions \cite{Gallager}
 \begin{subequations} \label{eqn:orthonormal-conditions}
 	\begin{align}
 		\langle \phi_i, \phi_q \rangle
@@ -82,9 +82,9 @@ Notice that assuming \(m_i\) and \(m_q\) are constant\footnote{This is an approx
 		&= m_i \underbrace{\int_T \phi_i \phi_i^* \,dt}_{1}
 			+ m_q \underbrace{\int_T \phi_q \phi_i^* \,dt}_{0} = m_i,
 \end{align*}
-which effectively means that it is possible to isolate a single component \(m_i(t)\) out of \(s(t)\). The same of course works with \(\phi_q\) as well resulting in \(\langle s, \phi_q \rangle = m_q\). Thus (remarkably) it is possible to send two signals on the same frequency, without them interfering with each other. Since each signal can represent one of \(\sqrt{M}\) values, by having two we obtain \(\sqrt{M} \cdot \sqrt{M} = M\) possible combinations.
+which effectively means that it is possible to isolate a single component \(m_i(t)\) out of \(s(t)\). The same of course works with \(\phi_q\) as well resulting in \(\langle s, \phi_q \rangle = m_q\). Thus (remarkably) it is possible to send two different signals over the same frequency, without them interfering with each other. Since each signal can represent one of \(\sqrt{M}\) values, by having two we obtain \(\sqrt{M} \cdot \sqrt{M} = M\) possible combinations.
 
-A graphical way to see what is happening, is to observe a so called \emph{constellation diagram}. An example is shown in \figref{fig:qam-constellation} for \(M = 16\). The two carrier signals \(\phi_i\) and \(\phi_q\) can be understood as bases of a coordinate system, in which the two amplitude levels of the two modulated messages, determine a position in the grid.
+A graphical way to see what is happening, is to observe a so called \emph{constellation diagram}. An example is shown in \figref{fig:qam-constellation} for \(M = 16\). The two carrier signals \(\phi_i\) and \(\phi_q\) can be understood as bases of a coordinate system, in which the two amplitude levels of the modulated messages determine a position in the grid.
 
 \paragraph{Example}
 
@@ -112,7 +112,7 @@ In \figref{fig:qam-constellation} the dots of the constellation have coordinates
 
 \subsection{Construction of orthogonal carrier signals}
 
-Knowing why there is a need for orthogonal carriers, we should now discuss which functions satisfy the property described by \eqref{eqn:orthogonal-condition}. If \(\phi_i\) is a real valued signal (which is typical) it is possible to find a function for the quadrature carrier using the \emph{Hilbert transform} (sometimes called Hilbert filter):
+Knowing why there is a need for orthogonal carriers, it should now be discussed which functions satisfy the property described by \eqref{eqn:orthogonal-condition}. If \(\phi_i\) is a real valued signal (which is typical) it is possible to find a function for the quadrature carrier using the \emph{Hilbert transform} (sometimes called Hilbert filter):
 \begin{equation}
 	\hilbert g(t) = g(t) * \frac{1}{\pi t}
 		= \frac{1}{\pi} \int_\mathbb{R} \frac{g(\tau)}{t - \tau} \,d\tau
@@ -126,27 +126,27 @@ The Hilbert transform is a linear operator that introduces a phase shift of \(\p
 % \subsection{Spectral properties of a QAM signal}
 % \skelpar[4]{Spectral properties of QAM}
 
-\section{Phase shift keying (\(M\)-PSK)}
+\section{Phase shift keying (\(M\)-PSK)} \label{sec:mpsk}
 
-Phase shift keying (PSK) is another popular family of modulation schemes for digital signals, that is simpler than QAM. In PSK as the name suggests only the phase of the envelope changes, which means that the symbols have all the same amplitude. Thus, instead of arranging the symbols into a grid as done in QAM, \(M\)-PSK distributes the symbols over the unit circle at equidistant intervals of \(2\pi / M\) radians \cite{Mathis,Kneubuehler}. An example of 8-PSK is shown in \figref{fig:psk-constellation}. Mathematically the process of a PSK modulation can be described by making the phase of a carrier function of the message signal. For a complex exponential carrier:
+Phase shift keying (PSK) is another popular family of modulation schemes for digital signals that is simpler than QAM. In PSK as the name suggests only the phase of the envelope changes, which implies that all symbols have the same amplitude. Thus, instead of arranging the symbols into a grid, as is done in QAM, \(M\)-PSK distributes the symbols over the unit circle at equidistant intervals of \(2\pi / M\) radians \cite{Mathis,Kneubuehler}. An example of 8-PSK is shown in \figref{fig:psk-constellation}. Mathematically the process of a PSK modulation can be described by making the carrier's phase of the message. For a complex exponential carrier:
 \begin{equation}
-	s(t) = \exp\left(\omega_c t + \varphi(t)\right), \quad\text{where}\quad
+	s(t) = \exp j\left(\omega_c t + \varphi(t)\right), \quad\text{where}\quad
 	\varphi = \frac{2\pi \cdot \text{Level}(\vec{m})}{M}, \quad \vec{m} \in \{0,1\}^{\log_2 M}.
 \end{equation}
 
-It is worth noting that the case of 4-PSK, also known as quaternary phase shift keying (QPSK), is a special case, because its constellation is (up to a constant phase) a 4-ary QAM.
+It is worth noting that the case of 4-PSK, also known as quaternary phase shift keying (QPSK), is a special case, because its constellation is (up to a constant phase difference) the same as a 4-ary QAM.
 
 \section{Multipath fading} \label{sec:multipath-fading}
 
 In the previous section, we discussed how the data is modulated and demodulated at the two ends of the transmission system. This section discusses what happens between the transmitter and receiver when the modulated passband signal is transmitted wirelessly.
 
-In theory because wireless transmission happens through electromagnetic radiation, to model a wireless channel one would need to solve Maxwell's equations for either the electric or magnetic field, however in practice that is not (analytically) possible. Instead what is typically done, is to model the impulse response of the channel using a geometrical or statistical model, parametrized by a set of coefficients that are either simulated or measured experimentally \cite{Gallager}.
+In theory because wireless transmission happens through electromagnetic radiation, to model a wireless channel one would need to solve Maxwell's equations for either the electric or magnetic field, however in practice that is usually not (analytically) possible. Instead what is typically done, is to model the impulse response of the channel using a geometrical or statistical model, that is parametrized by a set of coefficients, which are either simulated or measured experimentally \cite{Gallager}.
 
-In our relatively simple model we are going to include an additive white Gaussian noise (AWGN) and a Rician (or Rayleighan) fading; both are required to model physical effects of the real world. The former in particular is relevant today, as it mathematically describes dense urban environments.
+In our relatively simple model we are going to include an additive white Gaussian noise (AWGN) and a Rician (or Rayleighan) fading; both are required to model physical effects of the real world. The former in particular is relevant today, as it describes the noise patterns of dense urban environments \cite{Messier}.
 
 \subsection{Geometric model}
 
-The simplest way to understand multipath fading, is to consider it from a geometrical perspective. \figref{fig:multipath-sketch} is a sketch of a wireless transmission system affected by multipath fading. The sender's antenna radiates an electromagnetic wave in the direction of the receiver (red line), however even under the best circumstances a part of the signal is dispersed in other directions (blue lines).
+The simplest way to understand multipath fading, is to consider it from a geometrical perspective. \figref{fig:multipath-sketch} is a sketch of a wireless transmission system affected by multipath fading. The transmitter's antenna radiates an electromagnetic wave in the direction of the receiver (red line), however even under the best circumstances a part of the signal is dispersed in other directions (blue lines).
 
 \begin{figure}
 	\centering
@@ -157,14 +157,14 @@ The simplest way to understand multipath fading, is to consider it from a geomet
 	}
 \end{figure}
 
-The problem is that, as is geometrically evident, some paths are longer than others. Because of this fact, the signal is seen by the receiver multiple times with different phase shifts~\cite{Gallager,Messier}. To mathematically model this effect, we describe the received signal \(r(t)\) as a linear combination of delayed copies of the sent signal \(s(t)\), each with a different attenuation \(c_k\) and phase shift \(\tau_k\):
+The problem is that, as is geometrically evident, some paths are longer than others. Because of electromagnetic wave travel at a constant speed, the signal is seen by the receiver multiple times with different phase shifts~\cite{Gallager,Messier}. To analytically model this effect, we describe the received signal \(r(t)\) as a linear combination of delayed copies of the sent signal \(s(t)\), each with a different attenuation \(c_k\) and phase shift \(\tau_k\):
 \begin{equation} \label{eqn:geom-multipath-rx}
 	r(t) = \sum_k c_k s(t - \tau_k).
 \end{equation}
 
 The linearity of the model is justified by the assumption that the underlying electromagnetic waves behave linearly (superposition holds) \cite{Gallager}. How many copies of \(s(t)\) (usually referred to as ``taps'' or ``rays'') should be included in \eqref{eqn:geom-multipath-rx}, depends on the precision requirements of the model.
 
-A further complication arises, when one end (or both) is not stationary. In that case the lengths of the paths change over time, and as a result both the delays \(\tau_k\) as well as the attenuations \(c_k\) become functions of time: \(\tau_k(t)\) and \(c_k(t)\) respectively \cite{Gallager,Messier}. Even worse when the velocity at which the device is moving is high, then Doppler shifts of the electromagnetic wave frequency become non negligible \cite{Gallager}.
+A further complication arises, when one end (or both) is not stationary. In that case the lengths of the paths change over time, and as a result the delays \(\tau_k\) and the attenuations \(c_k\) become functions of time: \(\tau_k(t)\) and \(c_k(t)\) respectively \cite{Gallager,Messier}. Even worse is when the velocity at which the device is moving is high, since then Doppler shifts of the electromagnetic wave frequency become non negligible \cite{Gallager}.
 
 \begin{figure}
 	\centering
@@ -175,7 +175,7 @@ A further complication arises, when one end (or both) is not stationary. In that
 	}
 \end{figure}
 
-Thus the arrangement can be modelled as a linear time-\emph{varying} system (LTV), if the transmitter or the receiver (or anything else in the channel) is moving, and as a linear time \emph{invariant} (LTI) system if the geometry is stationary. Regardless of which of the two cases, linearity alone is sufficient to approximate the channel as a finite impulse response (FIR) filter \cite{Messier}. We can rewrite an LTV version of equation \eqref{eqn:geom-multipath-rx} using a convolution product as follows:
+Thus the arrangement can be modelled as a linear time-\emph{varying} system (LTV), if the transmitter or the receiver (or anything else in the channel) is moving, and as a linear time \emph{invariant} (LTI) system if the geometry is stationary. We can rewrite an LTV version of equation \eqref{eqn:geom-multipath-rx} using a convolution product as follows:
 \begin{align*}
 	r(t) = \sum_k c_k(t) s(t - \tau_k(t)) &= \sum_k c_k(t) \int_\mathbb{R} s(\tau) \delta(\tau - \tau_k(t)) \,d\tau \\
 		&= \int_\mathbb{R} s(\tau) \sum_k c_k(t) \delta(\tau - \tau_k(t)) \,d\tau = s(\tau) * h(\tau, t),
@@ -188,13 +188,13 @@ that describes the \emph{channel impulse response} (CIR). This function depends
 
 \subsection{Spectrum of a multipath fading channel}
 
-With a continuous time channel model the spectral properties of a fading channel can now be discussed, since the frequency response is the Fourier transform of the impulse response, i.e. \(H(f, t) = \fourier h(\tau, t)\). In this case \(h(\tau, t)\) depends on two time variables, but that is actually not an issue. It just means that the frequency response is also changing over time. Hence we perform the Fourier transform with respect to the channel (convolution) time variable \(\tau\) to obtain
+With a continuous time channel model the spectral properties of a fading channel can now be discussed. The frequency response is the Fourier transform of the impulse response, i.e. \(H(f, t) = \fourier h(\tau, t)\), though in this case \(h(\tau, t)\) depends on two time variables, but that is actually not an issue. It just means that the frequency response is also changing over time. Hence we perform the Fourier transform with respect to the channel (convolution) time variable \(\tau\) to obtain
 \begin{equation} \label{eqn:multipath-frequency-response}
 	H(f, t) = \int_\mathbb{R} \sum_k c_k(t) \delta(\tau - \tau_k(t)) e^{-2\pi jf\tau} \, d\tau
 	= \sum_k c_k(t) e^{-2\pi jf \tau_k(t)}.
 \end{equation}
 
-Equation \eqref{eqn:multipath-frequency-response} shows that the frequency response is a periodic complex exponential, which has some important implications. Notice that if there is only one tap (term), the magnitude of \(H(f, t)\) is a constant (with respect to \(f\)) since \(|e^{j\alpha f}| = 1\). This means that the channel attenuates all frequencies by the same amount, therefore it is said to be a \emph{frequency non-selective} or \emph{flat fading} channel. Whereas in the case when there is more than one tap, the taps interfere destructively at certain frequencies and the channel is called \emph{frequency selective}. To illustrate how this happens, plots of the frequency response of a two tap channel model are shown in \figref{fig:multipath-frequency-response-plots}. On the left is the magnitude of \(H(f, t)\), which presents periodic ``dips'', and on the right complex loci for the two taps (red and blue), as well as their sum (magenta), over the frequency range near the first dip (2 to 2.5 MHz) are shown.
+Equation \eqref{eqn:multipath-frequency-response} shows that the frequency response is a periodic complex exponential. This has some important implications. Notice that if there is only one tap (term), the magnitude of \(H(f, t)\) is a constant (with respect to \(f\)) since \(|e^{j\alpha f}| = 1\). This means that the channel attenuates all frequencies by the same amount, therefore it is said to be a \emph{frequency non-selective} or \emph{flat fading} channel. Whereas in the case when there is more than one tap, the taps interfere destructively at certain frequencies and the channel is called \emph{frequency selective}. To illustrate how this happens, plots of the frequency response of a two tap channel model are shown in \figref{fig:multipath-frequency-response-plots}. On the left is the magnitude of \(H(f, t)\), which presents periodic ``dips'' when the taps interfere destructively. On the right complex loci for the two taps (red and blue), as well as their sum (magenta) are shown, for values over the frequency range near the first dip (2 to 2.5 MHz).
 
 
 \begin{figure}
@@ -224,8 +224,7 @@ Another important parameter for quantifying dispersion is \emph{coherence bandwi
 	B_c \approx \frac{1}{T_d}.
 \end{equation}
 
-Finally, another important mean of parametrizing a multipath fading channel is what is called a \emph{power delay profile} (PDP). PDPs are nothing but a list of taps for a FIR model of multipath fading \cite{Mathis}. The weight of each tap in the PDP corresponds to the average channel tap power \(\expectation\{|h_l|^2\}\) (hence the name \emph{power} delay profile) and is usually given in decibel \cite{Mathis,Messier}. An example is shown at the end of chapter \ref{chp:implementation} in \tabref{tab:etsi-tap-values}.
-
+Finally, another important mean of parametrizing a multipath fading channel is what is called a \emph{power delay profile} (PDP). PDPs are nothing but a list of taps for a FIR model of multipath fading \cite{Mathis}. The weight of each tap in the PDP corresponds to the average channel tap power (hence the name \emph{power} delay profile) and is usually given in decibel \cite{Mathis,Messier}. An example is shown at the end of chapter \ref{chp:implementation} in \tabref{tab:etsi-tap-values}.
 
 % \subsection{Effects of multipath fading on modulation constellations}
 % 
@@ -238,7 +237,7 @@ Finally, another important mean of parametrizing a multipath fading channel is w
 
 % TODO: discuss the "bins" of discrete time
 
-Since in practice signal processing is done digitally, it is meaningful to discuss the properties of a discrete-time model. To keep the complexity of the model manageable some assumptions are necessary, thus the sent discrete signal\footnote{This is an abuse of notation. The argument \(n\) is used to mean the \(n\)-th digital sample of \(s\), whereas \(s(t)\) is used for the analog waveform. A more correct but longer notation is \(s(nT)\), where \(T\) is the sample time.} \(s(n)\) is assumed to have a finite single sided bandwidth \(W\). This implies that the time-domain the signal is a series of sinc-shaped pulses each shifted from the previous by a time interval \(T = 1 / (2W)\) (Nyquist rate) \cite{Messier}:
+Since in practice signal processing is done digitally, it is meaningful to discuss the properties of a discrete-time model. To keep the complexity of the model manageable some assumptions are necessary, thus the sent discrete signal\footnote{This is an abuse of notation. The argument \(n\) is used to mean the \(n\)-th digital sample of \(s\), whereas \(s(t)\) is used for the analog waveform. A more correct but longer notation is \(s(nT)\), where \(T\) is the sample time.} \(s(n)\) is assumed to have a finite single sided bandwidth \(W\) and the sampling is assumed to be ideal. This implies that the time-domain signal is a series of sinc-shaped pulses each shifted from the previous by a time interval \(T = 1 / (2W)\) (Nyquist rate) \cite{Messier}:
 \begin{equation}
 	s(t) = \sum_n s(n) \sinc \left(\frac{t}{T} - n\right).
 \end{equation}
@@ -264,7 +263,7 @@ This result is very similar to the continuous time model described by \eqref{eqn
 \begin{equation} \label{eqn:discrete-multipath-impulse-response}
 	h_l(m) = \sum_k c_k(mT) \sinc\left(l - \frac{\tau(mT)}{T}\right)
 \end{equation}
-is different from \eqref{eqn:multipath-impulse-response} consider again the plot of \(h(\tau,t)\) in \figref{fig:multipath-impulse-response}. The plot of \(h_l(m)\) would have discrete axes with \(m\) replacing \(t\) and \(l\) instead of \(\tau\), and because of the finite bandwidth in the \(l\) axis instead of Dirac deltas there would be superposed sinc functions.
+is different from \eqref{eqn:multipath-impulse-response} consider again the plot of \(h(\tau,t)\) in \figref{fig:multipath-impulse-response}. The plot of \(h_l(m)\) would have discrete axes with \(m\) replacing \(t\) and \(l\) instead of \(\tau\). Because of the finite bandwidth in the \(l\) axis instead of Dirac deltas there would be superposed sinc functions.
 
 \begin{figure}
 	\centering
@@ -275,11 +274,11 @@ is different from \eqref{eqn:multipath-impulse-response} consider again the plot
 	}
 \end{figure}
 
-From a signal processing perspective \eqref{eqn:discrete-multipath-impulse-response} can be interpreted as a simple tapped delay line \cite{Messier, Gallager}, schematically drawn in \figref{fig:tapped-delay-line}, which confirms that the presented mathematical model is indeed a FIR filter. Simple multipath channels can be simulated with just a few lines of code, for example the data for the static fading channel in \figref{fig:multipath-frequency-response-plots} is generated in just four lines of Python. The difficulty of fading channels in practice lies in the estimation of the constantly changing parameters \(c_k(t)\) and \(\tau_k(t)\) \cite{Messier}.
+From a signal processing perspective \eqref{eqn:discrete-multipath-impulse-response} can be interpreted as a simple tapped delay line \cite{Messier, Gallager}, as schematically drawn in \figref{fig:tapped-delay-line}. Together with linearity, this confirms that the presented model is indeed just a FIR filter. Simple multipath channels can be simulated with just a few lines of code, for example the data for the static fading channel in \figref{fig:multipath-frequency-response-plots} is generated in just four lines of Python. The difficulty of fading channels in practice lies in the estimation of the constantly changing parameters \(c_k(t)\) and \(\tau_k(t)\) \cite{Messier}.
 
 \subsection{Simulating multipath CIR with FIR filters} \label{sec:fractional-delay}
 
-As mentioned in the section before a FIR filter can be used to simulate discrete-time models of multipath fading. But with FIR filters the delays can only be integer multiples of the sample rate. When the delays are non integer an approximation needs to be done. That is because FIR filters have a transfer function of the form
+As mentioned in the previous section a FIR filter can be used to simulate discrete-time models of multipath fading. But with FIR filters the delays can only be integer multiples of the sample rate. When the delays are non integer an approximation needs to be done. That is because FIR filters have a transfer function of the form
 \begin{equation} \label{eqn:transfer-function-fir}
 	H(j\omega) = \sum_{n = 0}^{N} h(n) e^{-j\omega nT}
 	\quad \text{commonly written as} \quad
@@ -289,7 +288,7 @@ but a non integer delay of \(\tau\) in the frequency domain is \(H_\tau(j\omega)
 \begin{equation}
 	E(j\omega) = H(j\omega) - H_\tau(j\omega).
 \end{equation}
-The least square method plus the assumption of finite bandwidth and the requirement of causality gives the following rule for computing the FIR filter coefficients \cite{Valimaki1995}:
+The least square method together with the assumption of finite bandwidth and the requirement of causality gives the following rule for computing the FIR filter coefficients \cite{Valimaki1995}:
 \begin{equation}
 	h(n)= \begin{cases}
 		\sinc (n - \tau) & 0 \leq n \leq N \\ 
@@ -342,7 +341,7 @@ Equation \eqref{eqn:stat-wss} states that the fading CIR is a \emph{wide sense s
 
 Recall that \(h(\tau, t)\) is a function of time because \(c_k\) and \(\tau_k\) change over time. The idea of the statistical model is to replace the cumulative change caused by \(c_k\) and \(\tau_k\) (which are difficult to estimate) with a single random variable \(f\). This is done as follows.
 
-Multipath fading is a form of multiplicative noise, as mathematically confirmed by the fact that convolving a complex baseband signal \(e^{j\omega_c t}\) with the fading CIR \(h(\tau, t)\) gives
+Multipath fading is a form of multiplicative noise, as confirmed by the fact that convolving a complex baseband signal \(e^{j\omega_c t}\) with the fading CIR \(h(\tau, t)\) gives
 \begin{equation}
 	e^{j\omega_c \tau} * h(t, \tau) = \sum_k c_k(t) e^{j\omega_c(\tau - \tau_k(t))}
 	= e^{j\omega_c \tau} \sum_k c_k(t) e^{-j\omega_c \tau_k(t)}
@@ -355,7 +354,7 @@ If there is no line of sight (NLOS), it is reasonable to assume that all paths h
 \end{equation}
 where the \(c_k\) where omitted, since they are assumed to be all equal \cite{Hoher2013}. The factor \(1/\sqrt{N}\) is introduced such that \(\expectation \{|f|^2\} = 1\). It then can be shown that the probability density function of \(|f|\) is
 \begin{equation}
-	p(a)= 2a e^{-a^2}, \text{ or } |f| \sim \mathcal{R},
+	p(a)= 2a e^{-a^2}, \text{ or } |f| \sim \mathrm{Rayleigh},
 \end{equation}
 i.e. the amplitude of \(f\) is \emph{Raileigh} distributed \cite{Hoher2013}. The probability density function of a Rayleigh distributed random variable is shown in \figref{fig:rayleigh-rice-pdf}.